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THE LEAD WITH JAKE TAPPER
Plane Catches Fire on Tarmac; Debate Winners and Losers; Bush: My Campaign Is Not on Life Support. Aired 4-4:30p ET
JAKE TAPPER, CNN ANCHOR: Will last night turn out to be the last debate for anyone who was on that stage?
THE LEAD starts right now.
Who nailed it? Who face-planted? The answers to both of those questions, according to pundits, may be guys from the state of Florida, Marco Rubio riding high among the critics after a strong GOP debate. Could Jeb Bush be riding off into the sunset?
Plane on fire, terrifying video of a passenger jet spewing thick black smoke just seconds before takeoff, the latest on the injured and just what went wrong.
Plus, guns blazing outside a bar, the exclusive first dramatic look at the surveillance video of a bloody motorcycle gang shoot-out that left nine people dead and nearly 200 in jail.
Good afternoon everyone. Welcome to THE LEAD. I'm Jake Tapper.
Kicking off this Thursday with our politics lead. In what many pundits are calling Marco Rubio's big night, the Florida senator and ex-college football player spiked the ball during last night's GOP debate, per headlines and Republican insiders this morning, but the big questions, can Rubio turn what he's got into money, into a top poll ranking, into institutional support, into a real ground game in early states, into an actual thriving and viable presidential campaign that is in gear even when he's not out there charming a cable TV news camera?
I want to get right to CNN's Sunlen Serfaty. She's following Republican candidates on the campaign trail in Colorado.
Sunlen, fascinating last night, all eyes on Jeb Bush and yet it was Rubio who brought the jujitsu, using Jeb's attack against him.
SUNLEN SERFATY, CNN NATIONAL CORRESPONDENT: That's right, Jake. Jeb Bush, he really needed a big debate performance to breathe some life back into his campaign, so he attempted to take down Marco Rubio. It backfired on him and it really turned the night into Marco Rubio's moment.
(BEGIN VIDEOTAPE) SERFATY (voice-over): Tonight, new momentum for Marco Rubio.
SEN. MARCO RUBIO (R-FL), PRESIDENTIAL CANDIDATE: We can't afford to have another four years like the last eight years.
SERFATY: His strong debate performance could be a breakout moment for his campaign.
RUBIO: The Democrats have the ultimate super PAC. It's called the mainstream media.
SERFATY: But Rubio today trying to temper the hype.
RUBIO: Well, we feel good about it, as we did with the other two debates. The election wasn't decided last night. We are going to have another debate in 14 days. And that will replace some people's memory of this one.
SERFATY: Some damage control today from the Bush campaign after Rubio had the debate their candidate need.
UNIDENTIFIED MALE: Marco is obviously a very talented debater. Marco had some good zingers. In the broader debate over who do voters trust to do the job, Marco's been in the Senate for six years and he hasn't done anything.
SERFATY: Other struggling candidates did what they could to break through.
GOV. JOHN KASICH (R-OH), PRESIDENTIAL CANDIDATE: I want to tell you my great concern is that we are on the verge perhaps of picking someone who cannot do this job.
GOV. CHRIS CHRISTIE (R-NJ), PRESIDENTIAL CANDIDATE: We have $19 trillion in debt. We have people out of work. We have ISIS and al Qaeda attacking us and we're talking about fantasy football?
(CHEERING AND APPLAUSE)
SERFATY: Despite being center stage, the spotlight mostly avoided the front-runners. Donald Trump and Ben Carson seemed to play it safe.
DONALD TRUMP (R), PRESIDENTIAL CANDIDATE: Such a nasty question, but thank you, Governor.
KASICH: You're welcome.
SERFATY: Both fading into the background at times, providing an opening for another insurgent candidate, Ted Cruz, to stand out by taking on the moderators.
SEN. TED CRUZ (R-TX), PRESIDENTIAL CANDIDATE: How about talking about the substantive issues people care about?
SERFATY: Ben Carson today also taking aim at CNBC.
BEN CARSON (R), PRESIDENTIAL CANDIDATE: What it's turned into is a gotcha. That's silly.
SERFATY: And is trying to organize the other Republican candidates to push for changes to their next big encounter in less than two weeks.
CARSON: Debates are supposed to be established to help the people get to know the candidates and get to know what's behind them.
(END VIDEOTAPE)
SERFATY: So, clearly, Ben Carson today here in Colorado really hoping to capitalize on this moment of all this unhappiness over the debate, maybe even show a little leadership also, but, Jake, he did refuse to say whether he would boycott the next debate or not if those changes are not made.
TAPPER: All right, Sunlen Serfaty in Denver, Colorado, thanks so much.
Dr. Carson, of course, is the front-runner nationally in the latest "New York Times"/CBS News poll, though his lead over Donald Trump is within the margin of error.
Let's talk about last night's debate with Carson's adviser and business manager, Armstrong Williams.
Sir, thanks so much for being here. Appreciate it.
So, Dr. Carson today calling for a debate format that does not include gotcha questions. What does that mean? What is a gotcha question?
ARMSTRONG WILLIAMS, CARSON BUSINESS MANAGER: Well, you know, when you ask questions, when you're told that debate is about economics and you ask questions about gay marriage, which has nothing to do with the economy, you ask questions about fantasy football that has nothing to do with the economics of this country, you try to pit one candidate against the other, you know, those type of questions, Jake.
TAPPER: Well, one of the big moments, of course, in the debate last night that Dr. Carson objected to had to do with questions about his relationship with Mannatech, a company that does supplements, nutritional supplements. Was that a gotcha question?
WILLIAMS: No. That was a fair question.
TAPPER: That was a fair question?
WILLIAMS: Yes, absolutely.
TAPPER: I want to get back to that in a second. But let ask you just more broadly, what did Dr. Carson want to do last night, and did he achieve it? WILLIAMS: Well, Dr. Carson's -- the format, Jake, doesn't necessarily
play to his strength. So he's not going into the debate trying to be the winner and trying to dominate. That's not who he is, because he's not going to ever be disruptive to the rules. He's not going to ever overtalk the moderator and the other host who is on the panel.
TAPPER: It's not his nature to...
WILLIAMS: No, no. He abides by the rules. The rule says 30 seconds, he respects the rules. The rules say a minute, he respects them. He's not going to do what other candidates who are willing to insert, ignore the rules and guidelines. And it becomes very disruptive. He will never do that.
TAPPER: What do you think his appeal is, beyond obviously his personal charm and his life story? Why do you think Republican voters are responding to him so well?
WILLIAMS: Well, you may find this surprising. I think they feel that the media in some regards is unfair to him, no matter he's the front- runner. He wasn't treated as a front-runner last night at the debate. They tried to ignore him as much as they could because they wanted to highlight the other candidates to give them the momentum to try to close the gap between the two front-runners and the rest of the candidates.
People can see through that. And also Dr. Carson is given such a hard time for the gaffes that he's made. The media and others his don't seem to care about the intent and that he was well-meaning. So people feel that he's honest. They feel that he's sincere, he's trustworthy. And they're willing to give him a pass, because he doesn't have the oratory skills of a Carly Fiorina.
He will never have that. It's impossible. He's a surgeon, he's soft- spoken, he's a doctor, he has a quiet strength. People who know their pediatricians, that's exactly what they want. And just because you're the loudest person in the room doesn't mean that you're going to provide the best leadership. And the American people are willing to say, hey, you don't have to be the best on the debate stage, but when you're in my neighborhood, when you're in my church, when you're at my fair, you connect with me.
And they believe that he's sincere in what he says and they believe that he can make a difference.
TAPPER: Donald Trump obviously feeling the heat from the challenge from Dr. Carson, and even mentioned Dr. Carson's religion, saying Seventh Day Adventist, I don't know about that.
What did you make of that? Did you think that that was unfair? Do you take Trump on his word that he was merely saying he doesn't know anything about it? What did you think?
WILLIAMS: You know, we actually really respect and like Mr. Trump. We think he adds so much value to this debate in terms of issues that would not be discussed. I myself have criticized Mr. Trump when we thought he was being a
bully. But Mr. Trump showed his best Trump last night. We thought he was very respectful for Dr. Carson. He did not seem to show the insecurity and vulnerabilities of being of -- of leading from behind. And so we were surprised. But it was very encouraging for team Carson.
TAPPER: I want to settle this whole thing about Mannatech, because it came up last night and I do want to get some clarification. The question has to do with specifically this company that makes nutritional supplements.
And it's a firm that is now under scrutiny by state and federal officials for allegedly lying about the efficacy of benefits of the products. Here's what Dr. Carson said when he was asked about it.
(BEGIN VIDEO CLIP)
CARSON: Well, that's easy to answer. I didn't have an involvement with them. That is total propaganda, and this is what happens in our society. Total propaganda.
I did a couple of speeches for them. I do speeches for other people. They were paid speeches. It is absolutely absurd to say that I had any kind of a relationship with them.
Do I take the product? Yes. I think it's a good product.
(END VIDEO CLIP)
TAPPER: Now, here is a clip from a promotional video for Mannatech.
CARSON: It attracted me to the point that I actually began using those kind of products. I still use them to this very day. The wonderful thing about a company like Mannatech is that they recognize that when God made us, he gave us the right fuel.
TAPPER: And Dr. Carson said he wasn't paid for that, but that certainly seems to be a relationship. I have never done a testimonial like that for a product.
WILLIAMS: You know, I always understand that nothing is ever what it appears to be.
The good -- what is good about this is that I actually negotiated the contract...
TAPPER: OK.
WILLIAMS: ... as his business manager.
And Dr. Carson was asked to go to Arizona. And there's an e-mail chain -- and I may mess up some of the details -- to be a part of Mannatech. They wanted to tape something for PBS. But they were not sure that it was going to be nationally distributed until Dr. Carson actually did the event in Arizona.
Now, the script that they sent us that Dr. Carson was supposedly going to read from, when he actually got there to Arizona -- I was not with him -- he called. He said: "I'm not going to say this. I'm not comfortable with this. I don't believe in this."
And it created such disarray that -- I think the guy's name was Mr. Taylor (ph). He called me.
I said, look, he's not going to read the script. He thinks he's being taken advantage of. He doesn't want to disappoint the thousands of people that are in the audience. He's going to say what he wants to say. And I got to tell you, once he leaves Phoenix, this relationship is over, because part of the deal was, when we were negotiating, if it went well and PBS decided to distribute this, they would go back in March and tape part two.
There was like third -- part one, part two, part two. So Dr. Carson was never involved in the details. It's something myself and his executive secretary, R.G. Jones, were involved in. And once Dr. Carson realized the details, not us, he said: "I'm not comfortable with this. I can't do it."
TAPPER: But that does seem to suggest there was some relationship -- it's over now -- but there was some relationship at some point.
WILLIAMS: Well, obviously, when the Washington Speakers Bureau book you, Jake, to go and give a speech, sometimes, you may not know all the details, because there are some things that are negotiated that you're unaware of, like the script.
The script that we received was quite different from what was given to Dr. Carson in the prompter. And he just said: "I don't believe this. I'm not going to do it."
That was showing his integrity the fact that he said he wasn't going to -- and when that was over, he made it clear to me: "You need to get me out of this. I'm not going to do this again." And it was over.
TAPPER: Armstrong Williams, thank you so much for your time.
WILLIAMS: Thank you.
TAPPER: Really appreciate your helping to clear that up for us.
Coming up next, Jeb Bush in New Hampshire today after what people say was a whiff on a shot at Marco Rubio, laboring through another debate, he's insisting his campaign is doing just fine, it's not on life support. Should he go all in? Should he fold? More from the campaign trail next.
(COMMERCIAL BREAK)
[16:16:04] TAPPER: Welcome back to THE LEAD. I'm Jake Tapper.
Staying with our politics lead: in last night's Republican debate, Jeb Bush came in dead last, at least if you're judging the candidates on how many minutes they spoke. Many Bush supporters pulled up to their TVs last night looking for proof of life. And based on what we're hearing today, some of them didn't see it.
CNN senior political correspondent Brianna Keilar is in Portsmouth, New Hampshire.
Brianna, in the middle of the debate last night, Bush's campaign manager was seen confronting a CNBC administrator pushing for Jeb Bush to get more time. Now, today, Bush himself says his campaign is doing OK.
BRIANNA KEILAR, CNN SENIOR POLITICAL CORRESPONDENT: Yes, that's right.
And that frustration you saw there, Jake, we also saw a little bit of that from Jeb Bush out here on the campaign trail today. He said that he wished there had been more substantive questions. I think we also saw a humbled Jeb Bush because he said to the crowd here in Portsmouth, New Hampshire, you know, it was an acknowledgment that he didn't perform really well on the debate stage last night.
Even as you have Republicans who long wanted Jeb Bush to be their nominee they're now doubting him, Jeb Bush says he can stay in this. He says he has the money, the organization and perhaps most importantly the heart to see this campaign through.
(BEGIN VIDEOTAPE)
KEILAR (voice-over): Jeb Bush is back on the trail today, trying to connect with New Hampshire voters.
JEB BUSH (R), PRESIDENTIAL CANDIDATE: It's about fixing problems. It's about solving problems. It's about bringing people together, rather than tearing them apart. It's not about the big personalities on the stage. It's not about performance. It's about leadership. And the leader today in this country needs to be a unifier.
KEILAR: This after critics, including many Republicans, are panning his third debate performance.
BUSH: I'm running with heart. I'm not a performer. If they're looking for an entertainer in chief, I'm probably not the guy.
KEILAR: Bush struggled to make a mark and gave Marco Rubio a huge opening.
BUSH: Marco, when you signed up for this, this was a six-year term and you should be showing up to work. I mean, literally the Senate is it like a French workweek? You get three days where you have to show up? You can campaign. Or just resign and let someone else take the job.
SEN. MARCO RUBIO (R-FL), PRESIDENTIAL CANDIDATE: I don't remember you ever complaining about John McCain's vote record. The only reason you're doing it now is because we're running for the same position and someone has convinced you that attacking me is going to help you.
KEILAR: An exchange that must have stung even more coming from a man Bush mentored and helped get elected in the Senate.
BUSH: The most principled centered leader I know, Marco Rubio.
KEILAR: But five years and a presidential campaign later --
BUSH: His record of attendance was low prior to his announcement of his campaign. And I just think that's wrong.
KEILAR: Bush, the once presumed front runner who's raised $25 million for his campaign and is backed by a super PAC that's hauled in more than $100 million has had to cut payroll costs by 40 percent, and has dropped sharply in the polls.
BUSH: It's not on life support. We have the most money. We have the greatest organization. We're doing fine.
KEILAR: Now, political observers are widely concluding that Bush's presidential prospects are doomed even as he tries to project confidence that he can make a comeback.
BUSH: Look, there are two types of politicians. There are the talkers and there are the doers. I wish I could talk as well as some of the people on the stage, the big personalities on the stage. But I'm a doer.
KEILAR: Now, talking to Republicans who back Jeb Bush, they say this is why you can't write him off. He has campaign -- or he has money in reserves unlike any other campaign. Between his campaign and also between the super PAC that's backing him they've raked in $800 million.
But certainly I think there's an acknowledgment, Jake, that money isn't really enough and ultimately, he's going to have to have this breakthrough moment, something that really pushes him ahead of the pack. So, even those supporting him say he needs to make that happen but don't count him out yet.
TAPPER: Brianna Keilar, thanks so much.
Here now to break down the winners and losers of last night's Republican debate, CNN political commentators Amanda Carpenter and Ana Navarro.
Ana, Jeb Bush said it's about leadership, not debate performance.
[16:20:01] He said there are talkers and there are doers. But let's be honest, communication is a big part of the job, rallying
the country to your side, rallying Congress to do what you want. It seems a little empty to dismiss debates as just performance.
ANA NAVARRO, CNN POLITICAL COMMENTATOR: I agree with you. I think he needs to stop doing that. I think he needs to acknowledge that performance is part of the job description, embrace it and figure out how to tackle it.
The Jeb Bush I know and I've known him for over 25 years was an incredibly strong and inspiring governor. That's the guy I want to show up on that debate stage.
So, this idea that, you know, performing that entertaining is somehow selling out on your principles and your convictions and not being authentic is just false. It is part of the job description, particularly in 2015. You can't party like it's 1999. There's social media. There's blogs. There's Vines. There's Snapchat. There's all sorts of things which make debate performances that much more significant.
And they're being watched. Even that debate from hell last night, you know, badly moderated was watched by 14 million people, which is a huge number.
AMANDA CARPENTER, CNN POLITICAL COMMENTATOR: Yes. At one point --
TAPPER: I'm not going to the moderation of it. But, Amanda, what do you think about the Bush campaign at this point? You're someone who's plugged in. You've worked for Ted Cruz in the past. Do you think it's over for him?
Personally, I think it's very premature. He's got a lot of money in the bank. He's still a very accomplished governor.
CARPENTER: Well, at this point, maybe it's because Halloween is coming I feel like we're witnessing the walking Jeb. I mean, look at what Brianna said in the prepackage we were talking about where she says everyone acknowledges he has the most money, the most consultants, the most infrastructure, but somehow they are not translating all that manpower into a good candidate.
And it may be because Jeb I think he's getting bad advice. Whoever told him to go on that stage and attack Marco Rubio, it was almost a prewritten plan -- you must do this. And he did it and executed so poorly. He had bad line.
So, at some point, Jeb needs to take control of his campaign and figure out what his message is and what he's going to be about.
NAVARRO: Let me answer your question about whether I think Jeb is in a death spiral and the answer is flat no. Not only does he have the best team on the ground, not only is he rolling out policy, not only does he -- he also has the heart. He's one of the most fiercely competitive people I know, which is why it is incomprehensible to me that people ask me, does he have the heart? How is it that this man who I know to be a fierce competitor to have his heart in this full fledge is something not communicating that? And they have got 12 days, Jake, before the next debate.
TAPPER: FOX Business.
NAVARRO: They have got to tackle it. And he's got to slay this dragon dead.
TAPPER: Amanda, Ted Cruz for whom you used to work, other pundits say he had a strong night as well, especially his rant against the media. I guess the big question for Ted Cruz, can he turn this appeal to the Republican base broader? What's his ticket to appealing to suburban moms?
CARPENTER: Sure.
TAPPER: How does he do that?
CARPENTER: I mean, look at the hostility towards Washington right now goes across all party lines. There's nobody that has taken on the Washington machine, they call it the Washington cartel, as much in the last two years as Senator Ted Cruz.
I think he speaks to all people, Republican, Democrat, libertarian, progressive, whatever who are fed up with the way Washington does business. But more than that, aside from just anger at Washington, there's a whole side to him as a person, as a dad as someone who's gone through adversity in his life, and as a self-made man that people haven't had the chance to hear because he's been so embroiled in these policy battles.
And once he starts to, you know, unveil that side of him, people are going to be very surprised at the softer side of Ted Cruz.
NAVARRO: He needs to figure out how to speak to people like me. I've met him several times. Frankly the only thing I like about him is that he smokes Cuban cigars.
But I see him as an obstructionist. I see him as somebody that has not gotten anything done in the Senate other than promote his own persona as an obstructionist in the party.
So if he is going to grow the tent, he's got to figure out how he's going to deal with folks like Ana Navarro who see him that way.
TAPPER: One other question -- I know it's packed with liberals, but "The New York Times" editorial board calling on New Jersey Governor Chris Christie to drop out. They write, quote, "New Jersey is in real trouble and the governor is off pursuing a presidential run that's turned out to be noting more than a vanity project", according to the editorial.
Again, I know it's the "New York Times" editorial board but do they have a point?
CARPENTER: I think a lot of candidates need to drop out, starting with every person in the undercard debate. I mean, we've gone through several debates. People in the undercard who can't break above 1 percent in the polling, it's time to bid their candidacy adieu.
TAPPER: Really? In October?
CARPENTER: I think so. I mean, I give a lot of credit to retrospect to Governors Walker and Perry who dropped out when they really could have stayed in the race a lot longer for the good of the party, because they realize that the field needs to winnow.
[16:25:02] NAVARRO: Yes. But walker and Perry had something to lose. One had just finished being governor and the other one was a sitting governor.
Chris Christie has something to lose because his numbers in New Jersey are plummeting.
Some of the other guys in this under card debate, all they need is a plane ticket and sweater vest to show up to a debate. And so, it's very hard to make the argument that they should drop out.
TAPPER: Don't knock sweater vests.
Ana Navarro, Amanda Carpenter, thanks so much.
NAVARRO: I'm from Miami. I don't own any.
TAPPER: In our national lead, fire and smoke pouring out of a passenger plane seconds before takeoff. Several injuries reported. Why didn't the pilots know something was wrong?
Plus, bikers running and ducking, some pulling guns, returning fire during a shootout at a popular bar. That video is coming up.
TAPPER: Welcome back to THE LEAD.


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text:
Making 'Ink' with Jamin and Kiowa Winans
By Chris Jones on April 7, 2013 in Inspiration 0
Q – How did you come up with idea and then approach writing the script?
Jamin – When I was four, I always thought there was a monster in my room that looked like the witch from Snow White. My fear was that she would grab me out of bed and take me somewhere. That kept brewing in the back of my mind for all these years and around 2002 took that idea and fleshed it out a bit. Every six months or so I added a bit more to the concept and eventually I had an outline for Ink. That is my biggest part of the screenwriting process. I like to have a solid and extensive outline before I go to script. That way the writing goes quickly and I hopefully don't have to do many drafts. I usually outline extensively and like to work it out thoroughly before I get to script. I did this on Ink and then rewrote it about five times.
Q – Do you think character first or structure first?
Jamin – For me, so much of a story being interesting comes from understanding your character and how to get them into a conflict that's going to affect them the most. So character is first and foremost and then I built the structure around that.
Q – Who did you get script notes from and what were they like?
Jamin – Kiowa and another writing partner of ours gave me the script notes. The biggest thing we learned early on from the first draft was that there wasn't enough character development before things start taking off. Most standard structure will set up character first, get the audience familiar with them and then throw them into a world of conflict, but a lot of times I like to start on the conflict. I realized we needed to give the audience a little bit more to latch onto before we took them through this maze.
Q – Were you ever writing with the budget in mind?
Kiowa – Not really. Jamin was ambitious so the story could be the best it could be. It kind of seemed impossible at first, but we just made sure we had enough time to come up with a way to make it happen. We knew we weren't going to have enough money to throw at problems, in order to make it work.
Q – Once you locked the script, what was the next step?
Jamin – We hadn't intended on being really low budget. We always hoped we might get some real money. So the idea was that we'd try and get some name attached to it, but it's a horrible problem for indie filmmakers. You can't get a name without money and you can't get money without a name. So we just carried on and wrote a business plan to get private investment. We also wanted to invest in it, so we re-mortgaged our house. That was key because by being the first in with money, it was easier to get people we knew to part with theirs. The investors were more willing knowing that our money was on the line as well.
Kiowa – Then we set a date to start shooting. I think it was around September or October of 2006, we said we'd be shooting this film starting July 1st 2007. We didn't have any money, cast, or crew but just knew that we had to set a date and work backwards. So we started calling our crew and actors and doing fund raising at the same time. And guess what? We started shooting on that exact date. I feel that setting that kind of goal really lights a fire under you and you make it happen.
Q – What was your tipping point?
Kiowa –We live in a very small bungalow but for a lot of scenes in the movie we needed a big house. This way we could use it as multiple sets and a production office, so we decided to rent a large house in the suburbs. But the thing was we didn't sell our house, so by moving out we had double rent. Taking that bold step made us realize that we had to do it now and within two months of moving into that house, we had wrapped up our investment, hired everyone and were off to the races.
Jamin – Everyone committed at the same time. It was very apparent we were going to make the movie no matter what. I think if people feel that the film is going to happen with them or without them, no matter what, it makes a big difference. Plus we'd made a couple of films so they knew we were going to do it.
Q – How did you come up with the unique look of the film and how did you keep costs down?
Jamin – The key to keeping costs down was having a year of prep time. We started location scouting a year in advance in order to make sure we found the right forest with the leaves falling at the right time. We started working on the make up nine months in advance which was helpful because neither of us had ever done prosthetics before. So by the time we were really in production, we knew what we wanted to do and how we were going to do it. As for the look, let's start with Ink himself.
Remember, I wanted him to look like that old witch because one of the characters' main flaws is his vanity. So when he transforms into Ink, I wanted him to be hideous and have a very large nose. We worked with several make up artists in town and did tests so we knew how long the process was going take each day. As for how the fights looked, again, we took a long time choreographing them. We hired a very talented friend to be our fight choreographer. He had done various martial arts, wrestling expos and spent time as a bouncer at a bar so he's been in lots of fights. But the one thing he really knows is Parkour (free running)and we were able to put that into the fight sequences, which made them look fantastic.
Q – What about the color scheme and overall atmosphere?
Jamin – My cinematographer, Jeff, and I knew that with the many locations we were going to have there was a chance the audience might get disoriented. So we decided to make each one incredibly distinct visually so the audience would know where they are at every point. Giving them different color palettes in each world or dimension helped tremendously.
One key decision we made early on was that we wanted to shoot with a very small camera. We knew there would be an enormous amount of set-ups and shots and a lot of tight spaces, so we found the smallest HD camera we could find – the Sony-V1U. It allowed us to move quickly and if we needed to rig it to a car mount, a jib or a Steadicam, it could be done cheaply. We knew that would knock down our resolution a bit, but we did a test printing it to film and it worked really well.
This was also a very effects-heavy film. The most we've ever done and I was doing them myself. So we tested a lot there especially with the Storyteller's intro and exit into the world and the Incbui. Their look is a combination of visual and practical effects, so we had to do a lot to make sure they looked scary. And that was a good thing because on set they looked ridiculous. I thought I had made a horrible mistake. But it looked great and ironically the Incubi are the iconic images of the film.
Q – What program did you use to create their effects?
Jamin – Most of it was rotoscoping, which is why post took fourteen months. It was brutal. So for example, the Incbui have this screen in front of their faces with data moving across it. That was a combination of an actual plate of Plexiglas and then rotoscoped the effects in with After Effects. The glowing eyes are rotoscoping as well. It started with practical lighting on set and then I went in and animated them.
Q – Costuming was very elaborate in this film. How did you approach that?
Kiowa – We had three distinct things to create: the Storyteller's look, the Incbui's look and then Ink himself. Ink was very clear from the script, which says "He's wearing a cloak of rags that looks like an Apocalyptic bag lady." Very descriptive, but I also knew it was going to take a long time to make it. The Storytellers were harder because we had to come up with a concept outside of the script. So we put up pictures of things in magazines that inspired us. We knew going in that we wanted them all to look unique and conversely we wanted all the Incbui to look the same. So for the Incbui, we went sterile and plastic – where they are wearing rubber gloves up to their elbows, a plastic smock and this plate glass thing in front of their face. A lot of the look came because of cost as at one point we had to outfit a number of guys. But it probably made us be more creative. The Storytellers were my favorite part of the process because once we had the actors together I could design them individually. They are supposed to be angels that died at different points in history so their wardrobe reflected that. But the trick was they had to look cool individually as well as together so we couldn't overdo it. And then there was a practical issue, especially with the women. Because of all the fighting throughout the film, the costumes had to be such that everyone could move well or hide stunt padding – so no skirts!
Q – Did you go to thrift stores (charity shops) for these costumes?
Kiowa – Yes. But I had to go to regular stores too in order to get multiple sets of the costumes. If something got shredded or grape juice spilled on it without a back up, we would have been in a lot of trouble. I bought three exact replicas for each person. The troublesome one was the most elaborate – Ink. It took me nine months to create because it's a hand-sewn collage of old rags, sheets and curtains that I threw in the bathtub and dyed dark black. The shoulders of the cloak are hockey pads. I shredded it and did all sort of other things to make it look used. I also sewed other things onto it like the drums and shackles. It was the least functional costume to fight in. One particular drum would always hit him in the face! I had a tool belt on set every day with tons of safety pins and they would get so embedded in there, I would have to pull them out with pliers to make repairs and change the costume for continuity.
Q – How did you approach the art direction?
Kiowa – Jamin again had very specific ideas in mind, ideas that revolved around differentiating each world or dimension. For example, the Incbui hangout at the end had to be cold, sterile and somewhat modern because that's what they are. We built it in a warehouse and the background were different sized squares that were backlit plastic. I connected them with the two-inch binder rings and hoisted them up on wood frames. The scene with the bride was a set built in the garage of our "production" house, where we lived during the shoot. We bought every pink sheet and curtain in about every thrift store in town for that set. We had no money, so what we lacked in money we made up for in time.
Q – How did you go about getting locations?
Jamin – We got everything for free, except for the required permit to be in the forest. For 99% of our locations we just made a pact that said we wouldn't pay for one and if someone didn't give it to us for free, we moved on.
Kiowa – We looked for places that were already dressed. Props can become expensive especially if you only use them once. So the place where the Collector lives was this old warehouse with tons of metal. We knew we couldn't build that so we had to find a place that was ready to go.
Q – How did you organize the production?
Kiowa – It was mostly my computer and I. An important thing we did was set up our office in a room in the house we rented, which was also used as a set a number of times. I didn't have the luxury of a line producer so I did it. I figured out what our weekly cash burn was and worked to stay within that range. One thing I did to minimize the pain on my side was to set up automatic payments for various things like salaries. The actors I would have to pay individually because they didn't work all week long. But it worked well and also took pressure off my crew because I knew they didn't have the time to get to the bank. I didn't have time to print out checks and put stamps on them and people appreciated that we paid them on time and frequently.
Q – What about call sheets?
Kiowa – We had an AD/Production Manager who would create them based on our schedule. But in order to be efficient, I created a Flash website where all the PM had to do was drag a PDF call sheet document onto a server. The rule was all cast and crew had to check the website after 9PM where they could download the call sheet for the next day. It was a huge time saver.
Q – Did you get film insurance?
Kiowa – Yes. Liability and worker's comp (UK, Equipment, Public and Employers liability). You need that to rent equipment and get permits. We shot the whole thing in Denver and around Colorado. It's really not expensive. It was $300 for the liability and a little more for worker's comp. At least for the size film we were doing.
Q – Did you get film permits?
Kiowa – 90% of the time. We got to know the city engineer really well. It's very easy to shoot in Colorado. Film production isn't big business in Colorado so there isn't a lot of bureaucracy.
Jamin – The car accident scene was shot in the busiest downtown Denver intersection on a Sunday morning and we didn't pay anything for the permit. All we had to do was rent a few cops, get some barricades and pay for a traffic control plan. It cost about $2,200 in total but that's nothing!
Kiowa – The only time we got hassled by the cops was when we shot a fight scene in an alley. They were walking by and stopped. They wanted a picture with Ink. For the most part when shooting downtown we always got permits because we couldn't afford to lose a day.
Q – How many shooting days did you do?
Jamin – 83. 65 with our whole crew, which was around 10 people depending on the day. Most of them were on the grip and lighting crew. We also had stunt coordinators occasionally. That is where most of the budget went – salaries.
Q – What did you do about food?
Kiowa – It was not luxurious and we didn't have a dedicated craft service person. We had a cooler with water and energy drinks on which people got totally cracked out. We had a cooler of snacks – granola bars, etc. And then we would get lunch from a local place. It was different every day because we moved around so much. Food gets so expensive! In fact, we had so many people who wanted to come down and help, but we had to say no because we couldn't afford to feed them.
Q – How did you do the car wreck?
Jamin – A green screen sequence that was partly shot in our back driveway. The front shot had the camera inside on the passenger side of the Mercedes. We hung a green screen outside the driver side window and mimicked the wreck. We blew rubber glass and compressed air at our actor and then built an airbag that actually activated. Then we went to the actual intersection location and filmed a car hitting us. The way we did that was to have a car start at the car and go backwards, so when we played the footage in reverse it looks like the car is coming towards us. That was the back plate that went on the green screen. We did the same thing generally for the things that broke and reformed throughout the movie.
Q – Did you cut the film and if so in retrospect, would you rather have had someone else cut it?
Jamin – Yes, I did cut it. I have been an editor since I was a kid. It's the one part of the process I like and don't find stressful. It's always nice to work with a great editor and get their take on something, but in this case I had a very specific vision for the editing. So it was easier for me to do it. It took me about three to four months to get the first rough cut.
Q – What was your post-production like?
Kiowa – Well, I didn't know anything about post sound and yet I became the sound designer by default. It was by far the largest thing to tackle. It started with me just placing in the ambience. We recorded all the Foley ourselves. We made a sound booth in our basement by stapling up mattress foam. Jamin did all the color correction. We did everything over fourteen months, back to back, in our basement, over networked computers. We were so finicky because we knew we had to get it right. A lot of filmmakers don't do this because the experience has already been so painful up until that point. But there is so much you can do in post to make a project look polished. I'm glad we did it.
Q – Was it hard to work together even though you are married?
Kiowa – It was great! We work great together. We have the same tastes. And if he had gone through this stressful situation by himself, I wouldn't have been able to understand the why of it. But being part of it was really cool. There were very few occasions where we got fed up with one another. Also I have no idea how to be a director so there was no competition.
Jamin – It was unifying to go through it together and only solidified us, which is odd because stress doesn't always do that.
Q – What was your strategy with film festivals?
Jamin – We played at a lot of film festivals with our short films – over 80 – and they can and can't be helpful. For us, they didn't really help us. Having the shorts on YouTube did more. So when we were making Ink, we knew festivals wouldn't be our primary focus mostly because we didn't want it to play festivals for a year and become old news. To kick things off, we went to the first film festival that took us: Santa Barbara. It was close enough to LA to get that kind of exposure. We sent out postcards and e-mails to distributors. And one night an agent from UTA came to see it. Then the Ain't it Cool News review came out, which got us more attention. Then we opened theatrically in Denver.
Q – What happened there?
Jamin – It played for eight weeks when it was supposed to play for two. And then it went to two other theaters in Colorado where it played for an additional four weeks a piece. We brokered the deals with the theater owners ourselves. We have gotten offers for around $5,000-$10,000 from distributors, but we think the film is worth a lot more and know you'll never see any money outside of an advance. So we are hanging onto all the rights.
Kiowa – We decided to take the film out ourselves to independent theaters around the US. It's done well – at least 15 cities. But it's a lot of work. I have to show them our reviews, get them a screener and follow up. But we get a lot more people to see it this way and the buzz is better. And we earn some cash this way instead of wasting money sending formats and marketing materials to festivals so they can take your box office dollars. I think filmmakers have to embrace the fact that they are their own distributor due to the state of the indie market.
Q – Ink is now selling on DVD and iTunes to your fans, but it did get ripped and uploaded to torrent sites. How did that happen and how did it impact on your sales?
Jamin – We don't know who put the bit torrent up. We were told by someone very familiar with the file sharing world that as soon as Ink was available someone would definitely rip it within a day or two. Sure enough, it popped up right away. What we didn't expect was how it climbed to the top of Pirate Bay in about 48 hours. We certainly didn't push it, but we recognized it as a good thing. On IMDB, Ink went from a ranking of #12,991 to #16 in a week and then to #14 the following week. We were open with everyone that we were happy about it and that made news all over the piracy sites. I think the community appreciated our stance and thus came back to donate and support us. Pretty cool really.
Q – How do you avoid the trap of not working on your next project as you are so busy of the current one?
Jamin – Kiowa and I talk about that all the time. The way I get around it is that most of the promotional stuff falls to her. I spend some time on it, but mostly I am writing scripts, taking meetings, and mentally preparing for the next thing.
Q – What doors opened for you career-wise following 'Ink'?
Jamin – A lot of doors have opened. We're now represented by a large agency, UTA, and have had numerous meetings with studios and production companies that we never would have gotten before. However, because we intend to keep making independent films in which we have creative control, those meetings are less important than the fan base we've been building. Fans are always the most important thing to us. They're who we're making our films for and their support makes it possible for us to continue with or without Hollywood support. Ink has created a fan base we never thought possible.
Q – What advice would you give a new filmmaker?
Jamin – I think it's really important to be sure filmmaking is what you want to do with your life. Filmmaking is extraordinarily difficult, competitive, and painful contrary to the glamorous image it has. It's very hard to do and even harder to do well. You can't half-ass it, so be absolutely sure it's what you want to do. If you decide to pursue it then stay focused and persevere. It takes a lot of endurance and a lot of failure to have any successes. We're still learning that every day.
Great stuff, and a fantastic example of how costume can make a huge impact on a microbudget production.
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Pre-Oscar write up 2013
That time of year again. The worst part of my movie obsession comes to a thrilling conclusion as Hollywood honors the best films of the past year. Gives me something to talk about for a while. Alrighty, the game today is that I'll go thru the categories, ranking my picks and talking about them. By picks, I mean what I'd vote for, not necessarily what I think will win. Well, that's part of the discussion, but the ranks are based on my preference. I'll go as long as I can stand, meaning once my wrists hurt or my headache comes back or I stop caring about the rest of the categories, I'll stop. And here we go!
Argo - I have been team Argo from the first time I saw it. What impresses me most about this movie is the balance between suspense and comedy. I don't think I've ever seen a mix like that before, and I can't even pick which side of the film I prefer. Its truly original, and incredible overall.
Silver Linings Playbook - I'm picking this as my second choice mostly thinking of it in terms of being a "best picture", not necessarily my next favorite. I dont know, something just told me to put this movie next. I guess it just left me with some rather warm fuzzies. It's also one of the best ensembles put together in recent years. The first film in over 30 years to score nods in all four acting categories. That's gotta count for something
Django Unchained - My Tarantino obsession is no secret. I would love to see any film of his get this major recognition, although I really am more than happy with just the nomination. It may not be his best work, but even then it's much better than the majority of what Hollywood throws out there.
Les Miserables - This movie was truly beautiful. A few years ago, I may have gone with this as my top pick, but now it feels a little too obvious a choice. I like some quirk. But it's a brilliant adaptation of a fantastic story, backed up by a phenomenal cast.
Zero Dark Thirty - I love dark thrillers, and few have the weight that this one carries. Usually the genre is closer to guilty pleasure territory, but a film like this has real street cred. It's such an intense and intriguing ride. In a less competitive year, I'd have been more than happy to see it win. Well, I guess that's why we have The Hurt Locker, yeah?
Life of Pi - Another truly beuatiful film that would have been a shoo-in in another year. This film dazzled me and got me thinking. What seems like such a simple premise turns out to be such a wonderfully moving story. Wow, I can't believe I wrote something so sappy.
Lincoln - Yes, I know this is one of the top contenders this year. So why did I rank it so far down? Frankly, the movie is too perfect. Everything is top notch, but yet I don't find it particularly memorable or remarkable. Maybe it's because the expectations were so high, there was nothing to be surprised by. Or maybe there's something to be said for appreciating the flaws in a film.
Amour - This movie getting a nomination is a win in and of itself. I can see why it's struck such a strong chord with some members of the Academy. The fact that I didn't feel as strongly doesn't mean I can't appreciate it.
Beasts of the Southern Wild - This may be at a disadvantage for me because it's the only one I saw at home instead of in the theater. That does tend to diminish the experience for me a bit. I'm all in favor of a quirky indie getting some big recognition, but this just didn't resonate with me.
What's missing? - Of course, right after the nominations are announced, I have a really long list that now I can't recall. I had gotten excited about the possibility of Skyfall, so I was a lil sad that it didn't make the cut. Perks of Being a Wallflower was a big longshot that would have made me very happy. I wasn't as into Moonrise Kingdom as some others were, but I still sighed a lil when I noticed it didn't get in either.
Daniel Day-Lewis (Lincoln) - There's just no fighting this one. Had the nominations gone slightly different, I'd have voted elsewhere (we'll get there). But as much as I like the other actors in general, none of the other nominated performances come close to this one.
Hugh Jackman (Les Miserables) - Just taking on this role is worthy of a nomination. He carried a very difficult role with such ease. It's not an easy role to play, and certainly even more difficult to sing. This man had been long overdue for this sort of thing.
Bradley Cooper (Silver Linings Playbook) - The nomination is enough to make me happy to recognize this actor who's getting bigger and bigger with each film. Here we see so much more range and depth than we've gotten from Cooper before, and I'm excited to see what he does next.
Denzel Washington (Flight) - Such a truly incredible performance that no one's really talking about. There's such an intensity about it that's just heartbreaking to watch.
Joquin Phoenix (The Master) - I'm kinda upset with Phoenix for stealing this slot, especially given his ranting against awards. Sure, it was a good performance, but there are others I liked better who would have been more appreciative of the recognition.
Who's missing? - John Hawkes. John Hawkes. John Hawkes. That was the first glaring omission that truly upset me. I would have voted for him over DDL for his work in The Sessions. The physicality, the humor, the vulnerability, all of it just floored me. He provided such charm for the film, made it something truly special. I also would have _loved_ to see Jack Black for Bernie, but that was a long shot. Even longer would have been Jake Gyllenhaal for End of Watch. Tom Holland for The Impossible would have also been rather well deserved, as would Richard Gere for Arbitrage or Ben Affleck for Argo.
Jessica Chastain (zero Dark Thirty) - I was torn about how to rank these first two, but I hafta give it to Ms Chastain. The single moment that wins it for me is when she delivers this one line that should have been in a Tarantino film (I dont wanna spoil it). We see all sides of the emotional spectrum from her, and a fierce tenacity that's rare for a female role.
Jennifer Lawrence (Silver Linings Playbook) - The moment I realized I was actually torn about who I wanted to win was when I got excited at Lawrence's name being called for the Golden Globe. My first thought was "Wow, she is going to give an amazing acceptance speech" and she did, and for the few she's picked up since. I love this girl, and her spastic and clumsy humor is refreshing. She too has an intense role, making herself the strongest character in a movie full of A-listers.
Emmanuelle Riva (Amour) - I'm also torn on the ranking of these next two. But Riva had this realism that really hit me. I started reliving moments from my grandparents' final years. There was also this physicality to her role that was rather impressive.
Naomi Watts (The Impossible) - Also a breathtaking performance. Part of what lowers it on the list for me is that she does spend a lot of the film incapacitated and out of focus. But for the first half of the film, wow.
Quvenzhané Wallis(Beasts of the Southern Wild) - Impressive when you realize how young this girl was. She certainly proved she's a force to be reckoned with, but the nomination is plenty for this little one.
Who's Missing - I really wish more people would have seen Smashed because seeing Mary Elizabeth Winstead on this list would have made me so giddy. Sadly the film never got enough traction. I also thought Helen Mirren could have earned a spot for Hitchcock, but that film wasn't too well received. There was talk of Marion Cotillard for Rust and Bone, but personally, I wasn't that impressed with it.
Okay breaking the format a bit for this one. I dont feel as invested in this category because I have a list that's twice as long of actors I would have liked to have seen here. Part of what leaves me unimpressed by this lot is that all of them have won before. I know you technically should only vote for the nominated performance, but of course, there's far more factors at play. I guess I'd have to vote for Tommy Lee Jones in Lincoln, or Alan Arkin for Argo. Actually yeah, Arkin's more likeable. Him first. Next would be Robert DeNiro for Silver Linings Playbook, even though I dont think he should be rewarded simply for finally making a good movie this century. Then Christoph Waltz. I adore the man, but I would have preferred to see his costar Leonardo Dicaprio take his place. Last we've got Philip Seymour Hoffman. He's a favorite of mine, but I wasn't as big a fan of this movie.
And as for who's missing. My God, let's just list them, in no particular order: Leonardo Dicaprio (Django Unchained), Ewan McGregor (The Impossible), Matthew McConaughey (Magic Mike), Ezra Miller (Perks of Being a Wallflower), John Goodman (Argo). There were more, but that's all I can remember at the moment.
Anne Hathaway - Les Miserables - I know everyone's raving about her "I Dreamed a Dream" performance, but seriously, I had chills. I've loved this girl for a long time, and really wanted her to win for Rachel Getting Married (I'll spare you that long rant because it's pretty detailed). She's been so sweet and gracious every time she takes the stage to accept an award for this role, so she's gotta be able to do it one more time.
Helen Hunt (The Sessions) - Maybe not as strong a performance as her costar, John Hawkes, but still a wonderful and unique performance. Brave and corageous are the words that are being thrown around to describe it, and I have to concur.
Amy Adams (The Master) - I'm just biased and think that Amy Adams deserves an Oscar
Sally Field (Lincoln) - I have mixed feelings on this one. Yes it was a fantastic performance, and the bit where she puts down Tommy Lee Jones is worth her mention on this list. However, I still think she was miscast. She certainly was capable of delivering the performance, but she looked out of place. I'd hate to blame it on her age, but that's what I keep coming back to.
Jacki Weaver (Silver Linings Playbook) - She was just the most adorable thing ever in this movie, but she didn't have too much to do. Seems odd that she scored the nod. Didn't mean to rhyme. Apologies.
Who's Missing - Samantha Barks (Les Miserables), Ann Dowd (Compliance), and Emma Watson (Perks of Being a Wallflower). I dont think I'd have complained if any of them took any spot on this list (except for Hathaway's, of course)
Ang Lee (Life of Pi) - This movie could have easily crashed and burned, and that's all anyone would have talked about. But it didn't. Yet it's not getting much love for that acheivement. Water, animals, CGI, all very difficult elements to master on their own, and combining them is madness. Lee pulled it off masterfully.
Michael Haneke (Amour) - Really, I'd just love for the guy behind Funny Games to have Best Director on his resume.
David O Russell (Silver Linings Playbook) - Comedies (even if this one barely counts) don't often find themselves on this list. But Russell's nomination is very well deserved.
Steven Spielberg (Lincoln) - Okay, before I make my statement, I just want to say that I have nothing but respect for Spielberg, and he is one of the greatest filmmakers of all time. However, I feel like most of what he did for Lincoln was just be Spielberg. People want to work with him, so he brought in the best actors and writers and crew. Everyone is already so perfect, that it's hard to see his contributions to the film.
Ben Zeitlin (Beasts of the Southern Wild) - Oh that's nice.
Who's Missing - BEN AFFLECK! My gut reaction after the nominations was to first be incredibly upset over John Hawkes for actor, but as time went on, I got more and more upset over Affleck for Argo. What likely happened is that people thought he was a sure thing and gave their votes elsewhere. But if that's what guarantees Argo the win, then I'm fine with that. Also missing, well I'm never gonna pass up a chance to recognize Tarantino. Even if it was a tough year, he always has my support. Katheryn Bigelow was another glaring omission as was Tom Hooper.
Format's out the window for the rest of the post. All of this year's nominees are incredibly deserving: Django Unchained, Zero Dark Thirty, Flight, Moonrise Kingdom. Edge for me goes to Tarantino by default. I'm really sad that Looper didn't get on there. It's not easy to do time travel well, but that film was beyond impressive.
I'm still on team Argo, but I wouldn't mind letting this one go to Silver Linings Playbook. I'm mostly just upset that Perks of Being a Wallflower got snubbed. It had longshot potential in many categories, but this was it's best chance. Alas, it was shut out completely.
Not too much worth going into detail left. I love Wreck it Ralph for animated, and Anna Karenina for both creatives it's up for (costumes and production design). I'm kinda psyched that a Bond theme is poised to win song, even though it makes me ridiculously happy to say that Ted is an Academy Award nominated film. Sadly, I won't be able to watch the show on Sunday, but not too sad because I'll be arriving in Vegas to go to Eli Roth's Goretorium for my bday the next day. But this was a really strong year, and I'm excited to see the results
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In the news: Brett Butt's tough decision
April 10, 2008 Diane Wild 2 Comments
From the Canadian Press:
Upcoming season will be last for Corner Gas
"Brent Butt says the decision to pull the plug on his wildly popular and successful "Corner Gas" has left him feeling like something of a reluctant triggerman. "I feel like the kid at the end of `Old Yeller,'" Butt said Thursday after making a surprise announcement that the show, about the hijinks at a small-town Prairie gas station, would end its run at the end of its upcoming sixth season." Read more.
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2 thoughts on "In the news: Brett Butt's tough decision"
the regina mom says:
One has to wonder if the cuts the Govt of SK delivered to the film industry's training program figured into Butt's decision in any way…
I can't believe the show is ending. This is one of those shows the whole family can watch, and will have everyone in tears during the final episode, because there will be no more. This has to be one of the best shows I have seen since I was young, when tv was good. I am sorry to see this show ending, but I thank Brett and the cast for a wonderful time while watching. I'll be looking for the reruns. I feel like I'm losing family members. I hope this cast can get together in another show, as they blend together so well. Thanks folks! From an American viewer in Louisiana


text:
\section{Introduction}

Let $B(H)$ be the set of bounded linear operators acting on the Hilbert space
$H$ equipped with the inner product ${\langle} x, y{\rangle}$. If $H$ has dimension $n$,
then $H$ is identified with ${\mathbb C}^n$
with the inner product ${\langle} x, y{\rangle} = y^*x$, and
$B(H)$ is identified with the set $M_n$ of $n\times n$ complex matrices.
The numerical range of $T \in B(H)$ is defined as
$$W(T) = \{\langle Tx,x\rangle: x \in H, \langle x, x\rangle = 1\},$$
which is a useful concept in studying operators and matrices;
see, e.g., \cite{Halmos1,HJ}. In particular, there are interesting connections
between the study of numerical range and dilation theory; e.g., see
\cite{CL0,CL,Halmos2}.
Recall that an operator  $\hat T \in B(K)$
is a dilation of $T\in B(H)$ if $H \subseteq K$ and
$\hat T$ has an operator matrix of the form $\begin{pmatrix}
T & \star \cr \star & \star\cr\end{pmatrix}$ with respect to
a suitable orthonormal basis.
Equivalently, there is a partial isometry $X: H \rightarrow K$ with
$X^*X = I_H$ and $X^*\hat TX = T$.  We also say that $T$ is a compression
of $\hat T$.

\medskip
It is easy to show that $W(T) \subseteq W(\hat T)$ if $\hat T$
is a dilation of $T$.
Suppose  $T\in B(H)$ is a contraction, i.e., $\|T\| \le 1$. Then $T$
has a unitary dilation
$$U = \begin{pmatrix} T & \sqrt{I - TT^*}\cr \sqrt{I-T^*T} & -T^*\cr
\end{pmatrix} \in B(H \oplus H).$$
It was conjectured in \cite{Halmos2} that for any contraction $T \in B(H)$
\begin{equation}\label{classical}
W(T) = \cap \{ W(U): U \hbox{ is a unitary dilation of } T\}.
\end{equation}
However, counter-examples for the conjecture were given in \cite{D}.
In particular, there is a normal operator $T$ with $\|T\|\le 1$ such that
equality (\ref{classical}) fails. Denote by ${\bf cl}(X)$ the closure of a set in ${\mathbb C}$.
It was shown in \cite{CL} that for any contraction $T \in B(H)$,
$${\bf cl}(W(T)) = \cap \{ {\bf cl}(W(U)): U \hbox{ is a unitary dilation of } T\}.$$
This result has been refined, and extended to other types of generalized numerical
ranges; e.g., see \cite{BGT,BT,DM,GLW}.
In particular, the authors in \cite{DM} consider the extension of
the dilation result to the $C$-numerical range
defined as follows.
Let $C \in M_n$ and $T\in B(H)$ with $\dim H \ge n$.
Define the $C$-numerical range of  $T$ as
$$W_C(T) = \{ {\rm tr}\,(CX^*TX): X^*X = I_n\}.$$
For $H$ with $\dim H \ge n$,
if we regard $C \oplus {\bf 0}$ as a finite rank operator in $B(H)$, then
$$W_C(T) = \{{\rm tr}\,[(C \oplus {\bf 0})V^*TV]: V\in B(H) \hbox{ unitary}\}.$$
If $C$ is a rank one normal matrix with a nonzero eigenvalue $\gamma$ then
$W_C(T) = \gamma W(T)$;
if  $C = I_n\in M_n$ then $W_C(T)$ reduces to the
$n$-numerical range of $T$ consisting of complex numbers of the form
$\sum_{j=1}^n {\langle} Tx_j, x_j{\rangle}$ for an orthonormal set
$\{x_1, \dots, x_n\}\subseteq  H$.
If $C$ is normal with eigenvalues $c_1,\dots, c_n$, then
$W_C(T)$ consists of numbers of the form
$\sum_{j=1}^n c_j {\langle} Tx_j, x_j{\rangle}$ for an orthonormal set
$\{x_1, \dots, x_n\}\subseteq  H$, and the set is also referred to as
the $c$-numerical range  $W_c(T)$ of $T$ with $c = (c_1, \dots, c_n)$.
One may see \cite{GS,Li} and their references for some
basic background of the $C$-numerical range, and how it can be
used to study matrices and operators.

In \cite{DM}, the authors considered the extension of (\ref{classical})
to the $C$-numerical range. Examples of $C \in M_n$ and contraction $T \in B(H)$
with $\dim H \ge n$ are given such that
\label{WCA-i}
$${\bf cl}(W_C(T)) \ne \cap \{ {\bf cl}(W_C(U)): U \hbox{ is a unitary dilation of } T\}.$$
In this paper, we characterize  $C \in M_n$ such that the two sets above are
equal for any contraction $T \in B(H)$ with $\dim H \ge n$ by proving the following.

\begin{theorem} \label{key}
Let $C \in M_n$ be nonzero. The following conditions
are equivalent.
\begin{itemize}
\item[{\rm (a)}] The matrix $C$ is a rank one normal matrix.
\item[{\rm (b)}] For any contraction $T \in B(H)$ with $\dim H \ge n$,
\begin{equation}\label{intersection}
{\bf cl}(W_C(T)) = \cap  \{ {\bf cl} (W_C(U)): U \hbox{ is a unitary dilation of } T \}.
\end{equation}
\item[{\rm (c)}] For any contraction $T \in B(H)$ with $\dim H \ge n$,
\begin{equation}\label{closure}
{\bf conv}({\bf cl} (W_C(T))) =
{\bf conv} (\cap  \{{\bf cl} (W_C(U)): U \hbox{ is a unitary dilation of } T \}).
\end{equation}
\item[{\rm (d)}] For any rank one nilpotent contraction $T \in M_n$,
$$W_C(T) = \cap \{ W_C(U): U \hbox{ is a unitary dilation of }
T\}.$$
\end{itemize}
\end{theorem}

Several remarks concerning Theorem \ref{key} are in  order.

\begin{itemize}
\item
By Theorem \ref{key}, we see that (\ref{intersection}) holds for all contractions
$T \in B(H)$ with $\dim H \ge n$ if and only if
$C$ is a rank one normal matrix so that $W_C(A) = \gamma W(A)$ with $\gamma = {\rm tr}\, C$.

\item
In general, $W_C(T)$ may not be convex, and may have a complicated
geometrical shape. Condition (c) allows us to
characterize $C\in M_n$ satisfying
(\ref{intersection}) for all contractions
$T$ in terms of the convex hull of ${\bf cl}(W_C(T))$.

\item Condition (d) reduces the characterization of $C$ satisfying
(\ref{intersection}) or (\ref{closure}) for all contractions $T \in B(H)$
to a small test set, namely, the set of rank one nilpotent contractions in $M_n$.

\iffalse
\item
It is interesting to note that
$W_C(T)$ is convex if any one of the following holds;
e.g., see \cite{Li,LT,T,W} and their references.

* There is $\theta \in [0, \pi)$ such that $e^{i\theta} C$ is Hermitian.

* There is $\mu \in {\mathbb C}$ such that $C-\mu I_n$ has rank one.

* There is a unitary $V\in M_n$
and $\mu \in {\mathbb C}$
such that $V^*(C-\mu I_n)V =(C_{ij})_{1\le i,j \le k}$, where

 \quad $C_{jj} \in M_{n_j}$ for $j = 1, \dots, k$,  and
 only $C_{1,2}, \dots, C_{k-1,k}, C_{k,1}$ can be nonzero.

\fi
\item The implication (d) $\Rightarrow$ (a)
can be strengthened to the following.
 \end{itemize}

\begin{theorem} \label{main} If $C \in M_n$ is nonzero and is
not a rank one normal matrix, then there is a rank one nilpotent contraction
$T \in M_n$ such that $W_C(T)$ is a closed circular disk centered at the origin
with radius $r$, and there is a positive number $d>0$ such that
$r+d \in {\bf cl}(W_C(U))$  for any unitary dilation $U$ of $T$.
Consequently, $W_C(T) = {\bf cl}(W_C(T))$ is a proper subset of
\begin{equation}
\label{eq-fail}
\cap \{ {\bf cl}(W_C(U)): U\hbox{ a unitary dilation of } T\}.
\end{equation}
\end{theorem}

The proofs of Theorem \ref{key} and Theorem \ref{main}
will be presented in  the next section.
We note that the $C$-numerical range can be defined using
a trace class operator $C\in B(H_0)$, where $H_0$ is a separable
Hilbert space.  Our results and proofs
can be readily extended  a trace class operator
$C$ acting on a separable Hilbert space $H_0$ if
$M_n$ is replaced by $B(H_0)$  in the statements of
Theorem \ref{main} and Theorem \ref{key}.

\section{Proofs}

We need some auxiliary results to prove our theorems.
The following properties for the $C$-numerical range are known; see \cite{Li}.

\begin{itemize}
\item  Let $C, D \in M_n$ and $S,T \in B(H)$ with $\dim H \ge n$.
If $C = U^*DU$ and $S = V^*TV$
for some unitary $U, V$, then
$W_C(T) = W_D(S)$.

\item Let $C, A \in M_n$. Then $W_C(A) = W_A(C)$.

\item For $C \in M_n$ and $T \in B(H)$ with $\dim H \ge n$,
$W_C(\xi_1 I + \xi_2T) = \xi_1({\rm tr}\, C) + \xi_2 W_C(T)$.

\item Let $C \in M_n$. If $T \in B(H)$ is a compression of $\hat T$
and $\dim H \ge n$, then $W_C(T) \subseteq W_C(\hat T)$.
\end{itemize}

The following result will be used in our analysis;
See  \cite{S} and also \cite{Li,T} for its proof.

\begin{proposition} \label{prop1}
Let $A \in M_n$ be nonzero.
Then
$$R =
\min \{ \|A-\mu I\|: \mu \in {\mathbb C}\}
\quad \hbox{ is equal to }
\quad \max \{|x^*Ay|: x, y \in {\mathbb C}^n, \|x\| = \|y\| = 1, x^*y = 0\}.$$
Moreover,  $\mu \in {\mathbb C}$ satisfies $\|A-\mu I\| = R$ if and only if
$A-\mu I$ is unitarily similar to a matrix with the $(2,1)$ entry equal to
$R = \|A-\mu I\|$, which is the only nonzero entry
in the first column and the second row of the matrix $A-\mu I$.
\end{proposition}

\begin{lemma} \label{cor}
Suppose $C \in M_n$ and $T \in B(H)$ is a rank one nilpotent
operator with $\dim H \ge n \ge 2$. Let  $m = \min\{n+2, \dim H\}$
and $C_0 = C \oplus 0_{m-n} \in M_{m}$. Then
$$W_C(T) = \{ u^*C_0v: u,v \in {\mathbb C}^{m}, \|u\| = \|v\| = 1, u^*v  = 0\}$$
is a circular disk centered at the origin with radius
$r=\min\{ \|C_0-\mu I_m\|: \mu \in {\mathbb C}\}.$
\end{lemma}

\it Proof. \rm  Clearly,
$S = \{ x^*C_0y: x,y \in {\mathbb C}^{m}, \|x\| = \|y\| = 1\}
\subseteq W_C(T)$.
To prove the reverse inclusion,
let $\{x,y\}\subseteq H$ be an orthonormal set such that
$T(z) = \langle x,z\rangle y$, and
$\hat C_0 = C\oplus {\bf 0} \in B(H)$.
Suppose $V\in B(H)$ is unitary, and $\mu = {\rm tr}\,(\hat C_0VTV^*) =
\langle C_0 Vy, Vx\rangle  \in W_C(T)$.
If $\dim H \le n+2$, then  $\mu = u^*C_0v \in S$
with $(u,v) = (Vx, Vy)$.
If $\dim H  > n+2$, there is a unitary $U$
of the form $I_n \oplus U_1$ such that
$UV[x \ y] = [\tilde x \ \tilde y]$, and only the first $n+2$ entries of
$\tilde x$ and $\tilde y$ can be nonzero. Using the first $n+2$ entries for each
vector to form the
orthonormal pairs of vectors $u, v \in {\mathbb C}^{n+2}$, we see
that $\mu = u^*C_0v \in S$.

The description of the set $S$ follows from Proposition \ref{prop1}. \hfill $\Box$\medskip

The following is known as the elliptical range theorem for the
numerical range, see \cite{HJ,Li}. We list two special cases (a) and (b), which are
useful in our discussion.

\begin{lemma} \label{ellipse} Let $A = (a_{ij}) \in M_2$ with
eigenvalues $\lambda_1, \lambda_2$. Then $W(A)$ is an elliptical disk with
foci $\lambda_1,\lambda_2$, and length of minor axis
$\sqrt{{\rm tr}\, (A^*A)-|\lambda_1|^2 -|\lambda_2|^2}$.
\begin{itemize}
\item[{\rm (a)}] If $a_{11} = a_{22} = 0$,
then $W(A)$ is an elliptical disk with foci
$\pm \sqrt{a_{12}a_{21}}$ and length of minor axis equal to
$| |a_{12}|-|a_{21}| |$.
\item[{\rm (b)}] If $a_{21} = 0$, then
$W(A)$ is an elliptical disk with foci $a_{11}, a_{22}$, and
 length of minor axis equal to $|a_{12}|$.
\end{itemize}
\end{lemma}

We will often use the fact that $\mu \in W(T)$ for $T \in B(H)$ if and only if
there is a unitary operator $U\in B(H)$ such that $\mu$ is the $(1,1)$ entry
of $U^*TU$.
The following is a well-known fact about a boundary point of the
numerical range; see \cite{HJ,Li}.

\begin{lemma} \label{lemma}
Let $A=(a_{ij}) \in M_n$. If $a_{11}$ lies on the boundary of
$W(A)$, then  there is $\phi\in [0, 2\pi)$ such that
$e^{i\phi} a_{11} + e^{-i\phi} \bar a_{11}$ is
the largest eigenvalue of the Hermitian matrix
$A_\phi = e^{i\phi}A + e^{-i\phi}A^*$. Consequently,
the $(1,j)$ entry of $A_\phi$, i.e.,
$e^{i\phi} a_{1j} +  e^{-i\phi} \bar a_{j1} = 0$,
and hence
$|a_{1j}| = |a_{j1}|$ for $j = 2, \dots, n$.
\end{lemma}

Let $\{E_{11}, E_{12}, \dots, E_{nn}\}$ be the standard basis for $M_n$.
We have the following observation showing that for any unitary dilation
$U$ of the rank one dilation $T= \cos\theta E_{12} \in M_n$ with $\theta \in (0, \pi/2)$,
there is  a partial isometry $X: {\mathbb C}^{n+2} \rightarrow H$ such that
$X^*X = I_{n+2}$ and $X^*UX = \hat T \in M_{n+2}$ having some specific entries.
We can then use $W_C(\hat T)$ to establish our result.

\begin{lemma} \label{dilate}
Let $T = \cos \theta E_{12} \in M_n$ with  $n \ge 2$ and $\theta \in (0,\pi/2)$.
Suppose $U = \begin{pmatrix} T & * \cr * & * \cr \end{pmatrix} \in B(H)$
is a unitary dilation of  $T$. Then $\dim H \ge n+2$, and there is
a partial isometry $X: {\mathbb C}^{n+2} \rightarrow H$ such that
$X^*X = I_{n+2}$ and
\begin{equation}
\label{hatT}
\hat T  = X^*UX = \begin{pmatrix} T & T_{12}\cr T_{21} & T_{22}\cr\end{pmatrix}
\in M_{n+2},
\end{equation}
where
\begin{itemize}
\item the top two rows of $\hat T$ are
$(0, \cos \theta, 0, \dots, 0, -\sin \theta)$ and
$(0, \dots, 0, 1,0)$;

\item rows 3 to $n$ of $\hat T$  are zero;

\item the last two rows of  $\hat T$ equal

\medskip
\qquad $(x_1, x_2 \sin \theta, x_3, \dots, x_n, 0, x_2 \cos \theta)$ \ \ and \ \
$(y_1, y_2 \sin\theta, y_3, \dots, y_n, 0, y_2 \cos \theta)$,

\medskip\noindent
where
$x_4 = \cdots = x_n = 0$ if $n \ge 4$.

\end{itemize}
In particular, the
$4\times 4$ submatrix
of $\hat T$
with row and column indices $1,2,n+1, n+2$ equals
\begin{equation}\label{matrixB}
B = \begin{pmatrix}
0 & \cos \theta & 0 & -\sin \theta \cr
0 & 0 & 1 & 0\cr
x_1 & x_2\sin\theta & 0 & x_2\cos \theta\cr
y_1 & y_2\sin\theta & 0 & y_2\cos \theta\cr
\end{pmatrix}, \qquad \hbox{ where }
|x_1|^2 + |x_2|^2 \le 1, \ |y_1|^2 + |y_2|^2 \le 1.
\end{equation}
\iffalse
Consequently, if $C \in M_n$ with leading $2\times 2$ submatrix
$\hat C$, then
$W_{\hat C}(B) \subseteq W_C(U)$.
\fi
\end{lemma}

\it Proof. \rm
Since $U = \begin{pmatrix} T & \star \cr \star & \star\cr\end{pmatrix}$
is unitary, it has orthonormal rows. Thus,
there is a unitary operator $V = I_n \oplus V_1\in B(H)$  such that
the first two rows of $V^*UV$ have the form
$$(\underbrace{0, \cos\theta, 0, \dots, 0, 0, -\sin\theta}_{n+2}, {\bf 0})
\quad \hbox{ and }
\quad (\underbrace{0, \dots, 0, 1, 0}_{n+2}, {\bf 0}).$$
Evidently, if $H$ has dimension less than $n+2$, this is impossible.
Let $\hat T = \begin{pmatrix}T & T_{12} \cr T_{21} & T_{22}\cr\end{pmatrix}
\in M_{n+2}$ be the leading submatrix of  $V^*UV$.
Then the first two rows of $\hat T$
have the asserted form. Now the first two rows of $\hat T$ have unit length.
Thus, $\hat T \hat T^* = I_2 \oplus Y$ for some $Y \in M_n$.
So, all other rows of $\hat T$ are orthogonal to rows 1 and 2.
Since row 3 to row $n$ of $T$ are zero, we see that row 3 to row $n$
of $T_{12}$ must also be zero.
Also, the last two rows of $\hat T$ must have the form
$(x_1, x_2 \sin \theta, x_3, \dots, x_n, 0, x_2\cos\theta)$ and
$(y_1, y_2 \sin \theta, y_3, \dots, y_n, 0, y_2\cos\theta)$.
If $n \ge 3$, let $W_1\in M_{n-2}$ be unitary such that
$(x_3, \dots, x_n)W_1 = (\hat x_3, 0, \dots, 0)$.
Let $W = I_2 \oplus W_1 \oplus I_2$. We may replace
$\hat T$ by $W^*\hat T W$ and assume that
$x_4 = \cdots = x_n = 0$.
The assertion about $B$ is clear. \hfill $\Box$\medskip

\iffalse
For every matrix $X \in M_2$, there is a unitary $U \in M_2$
such that $U^*XU = (x_{ij})$ with $x_{11} = x_{22}$ and
$x_{21} \ge |x_{12}|$; see \cite{HJ}.
If  $C, A \in M_2$ are put in this special form,
then it was shown in \cite{Li2} (see also \cite{T})
that  $W_C(A)$ admits a nice description in the following.
\fi

\begin{lemma} \label{WCA}
Suppose $\theta \in [0, \pi/2)$, $f \in [0,1], g, x_2 \in {\mathbb C}$ with $|x_2| \le 1$,
\begin{equation} \label{matrixC}
\hat C  = \begin{pmatrix} g & f\cr 1 & g \cr\end{pmatrix} \quad\hbox{ and }\quad
\hat B = \begin{pmatrix} 0 &  1 \cr  x_2\sin\theta & 0 \end{pmatrix}.
\end{equation}
Let $M  = \begin{pmatrix} 0 & 2f x_2 \sin\theta \cr
2 & 0 \cr\end{pmatrix}$.
Then $W_{\hat C}(\hat B) = W(M)$
is the elliptical disk with foci $\pm 2\sqrt{ fx_2\sin\theta}$
and minor axis of length $2(1-f\sin\theta |x_2|)$.
Consequently, the intersection
of  $W_{\hat C}(\hat B)$ and the real axis always contains
the line segment  $[f\sin\theta-1, 1-f\sin \theta]$.
\end{lemma}

\it Proof. \rm
Since $\hat B$ is unitarily similar to
$\hat B^t$, we have $W_{\hat C}(\hat B) = W_{\hat C}(\hat B^t)$.
By the result in \cite{Li2} (see also \cite{Li,LT}),
$W_{\hat C}(\hat B^t) = W(M)$. By Lemma \ref{ellipse}
we have the description of $W(M)$. Thus, the intersection
of $W(M)$ and the real axis has the form $[-\xi, \xi]$.
Under the assumption $|x_2| \le 1$,
$\xi$ will attain the minimum value $\hat \xi = 1-f\sin\theta$
when $x_2 = -1$
so that the elliptical disk $W(M)$ has foci $\pm 2i\sqrt{f\sin\theta}$
and length of minor axis $2(1-f\sin\theta)$.
The last assertion follows.
\hfill $\Box$\medskip

Now, we are ready to present the proof of Theorem 1.2. We divide it into two proposition.

\begin{proposition} If $C$ is a scalar matrix, then Theorem {\rm \ref{main}} holds.
\end{proposition}

\it Proof. \rm
Suppose $C\in \gamma I_n$ with $\gamma \in {\mathbb C}$.
If $n = 1$, then $C$ is trivially a rank one normal
matrix. We assume $n \ge 2$ in the following.
We may replace $C$ by $C/\gamma$ and assume that $C = I_n$.
Let $T =  E_{12}/2\in M_n$.  By Lemma \ref{cor},
$W_C(T) = \{0\}$ if $\dim H = n$,
and $W_C(T)$ is a circular disk centered at 0 with radius $1/4$ if $\dim H > n$.

Suppose $U$ is a unitary dilation of $T$.
By Lemma \ref{dilate}, $U$ has a compression $\hat T\in M_{n+2}$
of form (\ref{hatT}), and
$\hat T$ has a principal submatrix $B$ of the form  (\ref{matrixB}).

{\bf Case 1.}
Suppose $|x_2| \le \sqrt 3/6$. By Lemma \ref{ellipse}, the submatrix
$B_0 = \begin{pmatrix} 0 & 1 \cr x_2 \sqrt{3}/2 & 0 \cr
\end{pmatrix}$ of $B$ has numerical range equal to an elliptical
disk centered at the origin
with the length of minor axis  $1 - |x_2|\sqrt 3/2 \ge 1 - 1/4 = 3/4$. Thus,
for any $\xi\in {\mathbb C}$ with $|\xi| \le 3/4$,
there is a unitary matrix $V = [1] \oplus V_1 \oplus [1]$ with $V_1 \in M_2$
such that $V^*BV$ has diagonal entries
$0, \xi, -\xi, y_2/2$.  Hence,
$\hat T$ is unitarily similar to a matrix $\hat T_\xi$ with diagonal
entries $0, \dots, 0, \xi, -\xi, y_2/\sqrt 2$ so that
$${\rm tr}\,((C\oplus 0_2)\hat T_\xi)
= {\rm tr}\,((I_n\oplus 0_2)\hat T_\xi) = \xi.$$
So, $W_C(U)$  contains
$\xi\in {\mathbb C}$ whenever $|\xi| \le 3/4$.

{\bf Case 2.} If $x_2 \ge \sqrt 3/6$, then  the submatrix
$\hat B_0 = \begin{pmatrix} 0 & x_2/2 \cr 0 & y_2 /2\cr
\end{pmatrix}$ of $B$ has numerical range equal to an elliptical disk
with foci $0, y_2/2$ and length of minor axis
$|x_2|/2 \ge \sqrt 3/12$. Thus, $0$ is an interior point of
$W(\hat B_0)$, and there is $d \in (0, 1/2)$ such that
$\xi \in W(\hat B_0)$ whenever $|\xi| \le d$.
Hence,
there is a unitary matrix $V = V_1 \oplus V_2$ with $V_1, V_2\in M_2$
such that $V^*BV$ has diagonal entries
$1/4,-1/4, d, y_2/2 -d$. Thus,
$\hat T$ is unitarily similar to a matrix $\hat T_d$ with diagonal
entries $0, \dots, 0, 1/4, d, -1/4, y_2/2 -d$, and
$${\rm tr}\,((C\oplus 0_2)\hat T_d)
= {\rm tr}\,((I_n\oplus 0_2)\hat T_d) =1/4 + d \in W_C(\hat T) \subseteq W_C(U).$$

\medskip
Combining Case 1 and Case 2, we see that $W_C(U)$ contains $1/4 + d$ for
every unitary dilation $U$ of $T$.
The conclusion of Theorem \ref{main} follows. \hfill $\Box$\medskip

\begin{proposition} If $C$ is not a scalar matrix, then Theorem {\rm (\ref{main})} holds.
\end{proposition}

\it Proof. \rm
Suppose $C$ is not a scalar matrix. Then $n \ge 2$ and
$R = \min\{\|C-\mu I\|: \mu \in {\mathbb C}\} > 0$.
By Proposition \ref{prop1}, we may
apply a suitable unitary similarity to $C$ and  assume that
$C$ has leading $2\times 2$ submatrix
$\begin{pmatrix} g & f \cr R & g\cr\end{pmatrix}$
with $R = \|C-gI\| = \min\{\|C-\mu I\|: \mu \in {\mathbb C}\}$. If $f = |f| e^{i\theta}$, we may
replace $C$ by $e^{-i\theta/2} D^*CD/R$ with $D = [e^{i\theta/2}]\oplus I_{n-1}$
and assume that the leading $2\times 2$ submatrix of $C$ is
$\hat C = \begin{pmatrix} g & f\cr 1 & g \cr \end{pmatrix}$.
Since $1 = \min\{ \|C-\mu I\|: \mu \in {\mathbb C}\} = \|C - gI\|$, we see that
$f \in [0,1]$,
the first column of $C$ has the form $(g,1,0,\dots,0)^t$ and
the second row of $C$ has the form $(1,g,0, \dots, 0)$.
We will prove the theorem under different assumptions on $f$ and $g$.

\medskip\noindent
{\bf (I)}
Suppose $1>f$. Let $T = \cos\theta E_{12} \in M_n$, where
$\theta \in [0, \pi/2)$ is sufficiently close to $\pi/2$, so that
$$1-f\sin \theta = \cos \theta + d  \quad \hbox{ with } d > 0.$$
By Lemma \ref{cor}, $W_C(T)$ is a circular disk
centered at the origin with radius $\cos\theta$.

\medskip
Suppose $U$ is a unitary dilation of $T$.
By Lemma \ref{dilate}, $U$ has a compression of the form $\hat T$
defined as in (\ref{hatT}).
The leading $2\times 2$ submatrix $\hat C$ of  $C$, and
the submatrix $\hat B$ of $\hat T$ in rows and columns
with indices 2 and $n+1$ have the form (\ref{matrixC}).

\smallskip
If $n = 2$, then we may assume that $C = \hat C$, and
$W_{\hat C}(\hat B) \subseteq W_C(\hat T) \subseteq W_C(U)$.
By Lemma \ref{WCA},  $[f\sin\theta -1, 1-f\sin\theta] \subseteq W_C(U)$.
Thus, $1-f\sin\theta = \cos \theta + d \in W_C(U)$.
Since this is true  for any unitary dilation $U$ of $T$,
the conclusion of Theorem \ref{main} follows.

Suppose $n \ge 3$.
Let $\tilde B = \begin{pmatrix} 0 & 1 & 0 \cr x_2 \sin\theta & 0 & x_3 \cr
 0 & 0 & 0 \cr\end{pmatrix}$. Then
$B_1 = \tilde B \oplus 0_{n-3} \in M_n$
can be obtained from  $\hat T$ by the deletion of the rows and
columns with indices $1, n+2$, followed by a permutation similarity.
Let $\tilde C =
\begin{pmatrix} g & f & c_{13} \cr
1 & g & 0 \cr
0 & c_{32} & c_{33} \cr\end{pmatrix}$
be the leading $3\times 3$ submatrix of $C$.
Suppose $\hat C$ and $\hat B$ defined in (\ref{matrixC}).
For any unitary $V = V_1 \oplus [\mu]$ with $V_1 \in M_2$,  let
$\xi_1 = {\rm tr}\,(\hat B V_1^*\hat C V_1)$ and $\xi_2 = (0, c_{32})V_1(0,x_3)^t$.
Then  the set of numbers
${\rm tr}\, (\tilde B V^*\tilde CV) = \xi_1 + \bar \mu \xi_2$
form a circle $S(V_1)$ with center $\xi_1$ and radius $|\xi_2|$.
By Lemma \ref{WCA}, we may choose $V_1$ such that
$\xi_1 = 1-f\sin\theta = \cos\theta + d$.
We will prove that $\xi_1 \in {\bf conv} S(V_1) \subseteq W_C(U)$.
To this end, we construct a continuous path
of unitary matrices $V_t \in [0,1]$ such that
$V_1$ is defined as above and $V_0 =\begin{pmatrix} 0 & 1 \cr 1 & 0 \cr\end{pmatrix}$.
For each $V_t$, let
$$S(V_t) = \{{\rm tr}\,(\tilde B(V_t\oplus [\mu])^* \tilde C (V_t \oplus [\mu])):
\mu \in {\mathbb C}, |\mu| =1\}\subseteq W_C(\hat T).$$
Let $\xi_1(t) =
{\rm tr}\,(\hat B V_t^*\hat C V_t)$ and
$\xi_2(t) = (0, c_{32})V_t(0,x_3)^t$.
Then $S(V_t)$ is a circle with center $\xi_1(t)$ and radius $|\xi_2(t)|$.
Note that $S(v_0) = \{ x_2\sin\theta + f\}$ is a singleton.
As $t$ varies from $1$ to 0, $S(V_1)$ will change to
the singleton $S(V_0)$ continuously. We see that
$\xi_1 \in {\bf conv} W(V_1) \subseteq W_C(\hat T) \subseteq W_C(U)$ as asserted.
We get the desired conclusion.

\medskip\noindent
{\bf (II)}
Suppose the leading $2\times 2$ submatrix $C$ equals
$\hat C = \begin{pmatrix} g & 1 \cr 1 & g\cr\end{pmatrix}$.
Since $C-gI_n$ has norm 1, which equals the $(1,2)$ entry and the $(2,1)$ entry
of the matrix. Thus, $C - gI_n$ is a direct sum of
its leading $2\times 2$ matrix and its trailing $(n-2)\times (n-2)$ submatrix.
As a result, $C = \hat C \oplus C_1$ with $\|C_1-g I_{n-2}\|\le 1$.

\medskip
Let $T = E_{12}/2$. By Lemma \ref{cor},
$W_C(T)=\{0\}$ if $\dim H = n$,
and $W_C(T)$ is a circular disk centered at 0 with radius $1/2$ if $\dim H > n$.
We will show that there is $d > 0$ such that $1/2 + d \in W_C(U)$
for any unitary dilation $U$ of $T$.

Suppose $U$ is a unitary dilation of $T$. By Lemma \ref{dilate},
$U$ has a compression $\hat T \in M_{n+2}$ of the form (\ref{hatT}),
and $\hat T$ has a principal submatrix
$B$ in the form (\ref{matrixB}) with $\theta = \pi/6$.

\medskip
{\bf (II.a)} Suppose $g \notin\{1,-1\}$.
Note that $\hat T$ is permutationally similar to a matrix of the form
$$\tilde T
= \begin{pmatrix} B &  \star \cr 0_{n-2,4} & 0_{n-2}\cr\end{pmatrix}.$$
So, for any unitary $V = V_1 \oplus I_{n-2} \in M_{n+2}$, where $V_1 \in M_4$,
$$\xi = {\rm tr}\,((0_2 \oplus C)V^*\tilde T V) =
{\rm tr}\,((0_2 \oplus \hat C)V_1^*BV_1)$$
is an element in $W_{\hat C}(B)$.
Also, every element in $W_{\hat C}(B)$ can be put in this form.

\medskip\noindent
{\bf Claim} \it If $B$ has the form $(\ref{matrixB})$,
then
$W_{\hat C}(B)$  contains an interval $[0, \xi]$ with $\xi > 1/2 = \cos\theta$.
\rm

\medskip
Suppose the claim holds.
Since the set  $\{(x_1, x_2, y_1, y_2)^t \in {\mathbb C}^4:
|x_1|^2 + |x_2|^2 \le 1, |y_1|^2 + |y_2|^2 \le 1\}$  is compact,
there is $d > 0$ such that  $W_{\hat C}(B)$ will contain a number larger
than $\cos\theta+d = 1/2 + d$ for any matrix $B$ in the form (\ref{matrixB}).
It will then follow that $1/2+d \in W_C(U)$ for any unitary dilation $U$ of $T$.

\medskip
To prove the {\bf claim},
let  $V = \frac{1}{\sqrt 2} \begin{pmatrix} 1 & 1 \cr 1 & -1 \cr\end{pmatrix}
\oplus I_2$,
and  $(c,s)  = (1/2,\sqrt 3/2)$. Then
$$\tilde B = V^*BV = \begin{pmatrix}
c/2   & -c/2 & 1/\sqrt{2} & -s/\sqrt{2}\cr
c/2   & -c/2 & -1/\sqrt{2} & -s/\sqrt{2}\cr
x_1/\sqrt{2} + x_2s/\sqrt{2} & x_1/\sqrt{2} - x_2s/\sqrt{2} & 0 & x_2c\cr
y_1/\sqrt{2} + y_2s/\sqrt{2}& y_1/\sqrt{2} - y_2s/\sqrt{2} & 0 & y_2c \cr
\end{pmatrix}.$$
For $j = 1,2$, let $B_j \in M_3$ be obtained from $\tilde B$ by deleting the
$j$th row and $j$th column so that
$$B_1 =  \begin{pmatrix}
 -c/2 & -1/\sqrt{2} & -s/\sqrt{2}\cr
 x_1/\sqrt{2} - x_2s/\sqrt{2} & 0 & x_2c\cr
 y_1/\sqrt{2} - y_2s/\sqrt{2} & 0 & y_2c \cr
\end{pmatrix} \ \hbox{ and } \
 B_2 = \begin{pmatrix}
c/2    & 1/\sqrt{2} & -s/\sqrt{2}\cr
x_1/\sqrt{2} + x_2s/\sqrt{2} & 0 & x_2c\cr
y_1/\sqrt{2} + y_2s/\sqrt{2} &  0 & y_2c \cr
\end{pmatrix}.$$
We consider two cases.

\medskip
{\bf Case 1} Suppose $-c/2$ is a boundary point of $W(B_1)$ and
$c/2$ is a boundary point of $W(B_2)$.
By Lemma \ref{lemma},
$$1 = |x_1 + x_2 s| = |x_1 - x_2 s| \quad \hbox{ and } \quad
s = |y_1 + y_2 s| = |y_1 - y_2 s|.$$
Thus, $0 =  x_1 \bar x_2 + \bar x_1 x_2, 1 = |x_1|^2 + s^2|x_2|^2$.
Since $|x_1|^2 + |x_2|^2 \le 1, |y_1|^2 + |y_2^2| \le 1$, we see that
$x_2 = 0$ and $|x_1| = 1$.
Since $|x_1|^2 + |y_1|^2 \le 1$, it follows that $y_1 = 0$, and
 $|y_2s| = s$, i.e., $|y_2| = 1$.
But then the matrix $\hat B$ in (\ref{matrixC})
will be of the form
$\begin{pmatrix} 0 & 1 \cr 0 & 0 \cr\end{pmatrix}$
so that $W_{\hat C}(\hat B) = W_{\hat C}(\hat B^t) = W(M)$,
where $M = \begin{pmatrix} 0 & 0 \cr 2 & 0 \cr\end{pmatrix}$ by
Lemma \ref{WCA}. Thus,  $W_{\hat C}(\hat B)$
 is the unit disk containing the interval $[0,1]$.

{\bf Case 2} Suppose
$-c/2$ is an interior point of $W(B_1)$ or
$c/2$ is an interior point of $W(B_2)$.
Here recall that we assume that $g \notin \{1, -1\}$.
If $-c/2$ is an interior point of $B_1$, then there is $\delta > 0$ such that
$-c/2 + \varepsilon/(g-1) \in W(B_1)$ for any $|\varepsilon|< \delta$, and
$\tilde B$ is unitarily similar to a matrix with its leading
$2\times 2$ submatrix
$$B_3 = \begin{pmatrix} c/2 & \star \cr
\star & -c/2+\varepsilon/(g-1)\cr\end{pmatrix}.$$
The matrix $\hat C = \begin{pmatrix} g& 1 \cr 1 & g \cr\end{pmatrix}$
is unitarily similar to
$\tilde C = {\rm diag}\,(g+1, g-1)$.
Thus, $W_{\tilde C}(B_3)$ contains real numbers of the form
$(g+1)c/2 + (g-1)(-c/2 + \varepsilon/(g-1)) = c + \varepsilon$.
Hence,  $W_{\tilde C}(B_3)$ contains the interval $[c, c+\varepsilon]$.

\medskip
Similarly, if $c/2$ is an interior point of $B_2$, then there is
$\delta > 0$ such that
$c/2 + \varepsilon/(g+1) \in W(B_2)$ whenever $|\varepsilon| < \delta$, and
$\tilde B$ is unitarily similar to a matrix with its leading
$2\times 2$ submatrix
$$B_4 = \begin{pmatrix} c/2+\varepsilon/(g+1) & \star \cr
\star & -c/2\cr\end{pmatrix}.$$
Then, $W_{\tilde C}(B_4)$ contains  real numbers of the form
$(g+1)(c/2 + \varepsilon/(g+1)) - (g-1)c/2 = c + \varepsilon$.
Hence,  $W_{\tilde C}(B_4)$ contains the interval $[c, c+\varepsilon]$.

\medskip
Combining the above two cases, we establish the {\bf claim}. The  theorem follows.

\medskip
{\bf (II.b)} Suppose $g \in \{1, -1\}$. Then $\hat C$ is  unitarily similar to
${\rm diag}\,(2g,0)$.
We may assume that $g = 1$. Otherwise, replace $C$ by $-C$.
Thus, $W_{\hat C}(B) = 2W(B)$.
Since $C$ is not a rank one normal matrix, $C_1 \ne 0$.
Thus, $C_1$ is unitarily similar to a matrix with a nonzero $(1,1)$ entry
$\mu$.

{\bf Case 1.}
Suppose in the matrix $B$,
$|x_2| \le \sqrt 3/6$. By Lemma \ref{ellipse}, the submatrix
$B_0 = \begin{pmatrix} 0 & 1 \cr x_2 \sqrt{3}/2 & 0 \cr
\end{pmatrix}$ of $B$ has numerical range equal to an elliptical
with the length of minor axis  $1 - |x_2|\sqrt 3/6 \ge 1 - 1/4 = 3/4$. Thus,
there is a unitary matrix $V = [1] \oplus V_1 \oplus [1]$ with $V_1 \in M_2$
such that $V^*BV$ has diagonal entries
$ 0, \xi,  -\xi, y_2/2$ for any $|\xi| \le \sqrt 3/4$, and
$\hat T$ is unitarily similar to a matrix of the form
$\hat T_\xi = \begin{pmatrix}
V^*BV & \star  \cr 0_{n-2,4} & 0_{n-2} \cr \end{pmatrix}$.
Since  $C\oplus 0_2$ is unitarily similar to
$\tilde C = {\rm diag}\,(0,2,0,0) \oplus C_1$,
${\rm tr}\,(\tilde C \hat T_\xi) = \xi.$ So, $W_C(U)$ always contains
$\xi$ with $|\xi| \le \sqrt 3/4$.

{\bf Case 2.} If $x_2 \ge \sqrt 3/6$, then  the submatrix
$\hat B_0 = \begin{pmatrix} 0 & x_2/2 \cr 0 & y_2 /2\cr
\end{pmatrix}$ of $B$ has numerical range equal to an elliptical
with foci $0, y_2/2$ and length of minor axis
$|x_2|/2 \ge \sqrt 3/12$. Thus, there is $\delta > 0$ such that
$\xi \in W(\hat B_0)$ whenever $|\xi| < \delta$.
Thus, for any $\xi_1,\xi_2$ with $|\xi_1| \le 1/4$ and $|\xi_2| \le \delta$,
there is a unitary matrix $V = V_1 \oplus V_2$ with $V_1, V_2\in M_2$
such that $V^*BV$ has diagonal entries
$\xi_1,-\xi_1, \xi_2, y_2/2 -\xi_2$,
$\hat T$ is unitarily similar to a matrix of the form
$\hat T_\xi = \begin{pmatrix}
V^*BV & \star \cr 0_{n-2,4} & 0_{n-2} \cr
\end{pmatrix}$.
Since, $C\oplus 0_2$ is unitarily similar to
$\tilde C = {\rm diag}\,(2,0,0,0) \oplus C_1$, we have
${\rm tr}\,(\tilde C \hat T_\xi) = \xi_1 + \mu \xi_2.$ Hence,
$W_C(U)$ always contains $\xi$ with $\xi \in (0, 1/4 + |\mu|\delta]$.

\medskip
Combining the two cases, we see that
$W_C(U)$ always contains $1/2 + d = \min\{\sqrt 3/4, 1/4 + |\mu|\delta\}$.
The  theorem follows.
\hfill $\Box$\medskip

By the above two propositions, Theorem \ref{main} holds. Next we turn to following.

\medskip
\noindent
{\bf Proof of Theorem \ref{key}}
The implication (a) $\Rightarrow$ (b) follows from the result
in \cite{CL}. If (b) holds, then (c) clearly holds.
Suppose (c) holds. By Lemma \ref{cor}, $W_C(T)$ is a closed circular disk
centered at the origin. Hence
\begin{eqnarray*}
W_C(T)
&\subseteq& \cap \{W_C(U): U \hbox{ is a unitary dilation of } T\} \\
&\subseteq& {\bf conv}(\cap \{{\bf cl}(W_C(U)): U \hbox{ is a unitary dilation of } T\})\\
& = & {\bf conv}({\bf cl}(W_C(T))) \hskip .5in \hbox{  by   (c)} \\
& = & W_T(C)  \hskip 1.2in \hbox { since $C$ is a closed circular disk}.
\end{eqnarray*}
We get condition (d).

By Theorem \ref{main}, if (a) does not holds, then
(d) does not hold. Thus, we have the implication
(d) $\Rightarrow$ (a). \hfill $\Box$\medskip

\medskip\noindent
{\bf Acknowledgment}

The author would like to thank the referee for
her/his careful reading of the manuscript, and helpful comments.

\medskip\noindent
{\bf Declaration.}

There is no conflict of interest connected to this article.

redpajama_set_name: RedPajamaArXiv


text:
Die Campiña de Jerez ist eine Comarca in der spanischen Provinz Cádiz. Sie wurde, wie alle Comarcas in der Autonomen Gemeinschaft Andalusien, mit Wirkung zum 28. März 2003 eingerichtet.

Geographie

Die Comarca liegt am Guadalete in den Montes de Propio und teilweise im Naturpark Los Alcornocales.

Gemeinden
Beide Gemeinden der Comarca bilden auch die Mancomunidad de Municipios Bahía de Cádiz und sind Teil der Metropolregion Bahía de Cádiz-Jerez.

Wirtschaft
Wichtigster Wirtschaftszweig ist die Landwirtschaft. Neben Wein im Dreieck Jerez de la Frontera-Sanlúcar de Barrameda-El Puerto de Santa María werden auch Baumwolle und Rote Rüben angebaut.

Galerie

Siehe auch
 Liste der Comarcas in Spanien

Einzelnachweise

Comarca in der Provinz Cádiz
Jerez de la Frontera

redpajama_set_name: RedPajamaWikipedia