text:
\section{Introduction}
It is well-known that the spectrum of a selfadjoint operator $A$ in
a Hilbert space ${\mathcal H}$ can be approximated by the spectra of a
sequence $(A_n)$ of selfadjoint operators in ${\mathcal H}$ converging to $A$
in strong resolvent sense. This fact--sometimes called
semi-continuity of the spectrum--implies that the spectrum cannot
suddenly expand in the limit. Applied to a family of equivariant
selfadjoint operators over a minimal dynamical system this
semi-continuity yields that the spectrum (as a set) agrees for all
operators in that family, see \cite{Lenz2, Joh, CFKS, BIST,
LenzSeifertStollmann2014} for various versions of this theorem.
However, semi-continuity of the spectrum fails to be true for
non-selfadjoint operators \cite[Example IV.3.8]{Kato1980}.

Nevertheless, as shown in \cite{BeckusLenzLindnerSeifert2015}
constancy of the spectrum is still valid for suitable families of
operators in $\ell^p ({\mathbb Z})$ which are equivariant over dynamical
systems with a minimal ${\mathbb Z}$ action, and \cite{Seifert2015} generalizes this to the case $\ell^p({\mathbb Z}^N)$.
This suggests  that minimality
of the dynamical system $(\Omega,{\mathcal G},\alpha)$, where $\Omega$ is a
compact metric space, ${\mathcal G}$ is a countable group and $\alpha$ is
the group action of ${\mathcal G}$ on $\Omega$, implies constancy of the
spectrum of an equivariant family of (not nesessarily selfadjoint)
operators  $(A(\omega))_{\omega\in\Omega}$ on ${\mathcal G}$. Indeed,  as far
as dynamical systems are concerned the arguments given in
\cite{BeckusLenzLindnerSeifert2015, Seifert2015} are rather general (and simple)
and not restricted to  ${\mathbb Z}$ and  it was already noted in the
introduction of \cite{BeckusLenzLindnerSeifert2015} that the results
can be extended to all situations where a suitable theory of limit
operators is at hand.

Here, we discuss how such a suitable theory of limit operator is
indeed at hand in a rather general situation dealing with Banach
space valued $\ell^p$ spaces. As a consequence we obtain  constancy
of the spectra  for ${\mathcal G}$ being countable discrete (not nesessarily abelian) and
$(A(\omega))_{\omega\in\Omega}$ is a (suitable) family of bounded
operators on $\ell^p({\mathcal G},Y)$, where $p\in[1,\infty]$ and $Y$ is
Banach space. In order to prove this fact we make use of the theory
of limit operators developed in
\cite{RabinovichRochSilbermann1998,RabinovichRochSilbermann2004,Lindner2006,CL08,CL10,RaRoRoe,LindnerSeidel2014,SpakWillett}.
The right topological framework turns out to be the so-called
${\mathcal P}$-theory, see
\cite{RochSilbermann1989,ProssdorfSilbermann,RabinovichRochSilbermann2004,Seidel2014}.
Following  \cite{BeckusLenzLindnerSeifert2015} we introduce the
notion of pseudoergodic elements of $\Omega$, as first appeared in
\cite{Davies2001b} in a somewhat different way. We will show that
pseudoergodicity of elements is closely related to minimality of the
dynamical system. However, for pseudoergodic elements the set of
limit operators can be described rather easily, thus yielding the
assertion of our main theorem. We moreover prove constancy of the
pseudospectrum, which is an intensively studied object for
non-selfadjoint operators, see \cite{TrefethenEmbree2005} and
references therein.

In Section \ref{sec:P-theory} we collect basic notions and facts
from the ${\mathcal P}$-theory needed for our purpose. We will also introduce
the class of operators on groups we consider. Section
\ref{sec:dynamical_systems} deals with dynamical systems and shows
the interplay between pseudoergodicity of elements and minimality.
We will combine these two things in Section
\ref{sec:operator_over_DS} when we introduce the families of
operators over dynamical systems. There, we also state and prove our
main theorems. In Section~\ref{sec:operators_over_finitely_generated_groups} and Section~\ref{sec:operators_over_ZN} we specialize to the case of finitely
generated groups and to the group ${\mathbb Z}^N$, which is mostly dealt
with in the literature
\cite{RabinovichRochSilbermann1998,RabinovichRochSilbermann2004,Lindner2006,CL10,LindnerSeidel2014}.

\section{Limit operators, the ${\mathcal P}$-theory and operators on discrete groups}
\label{sec:P-theory}

In this section we recall the ${\mathcal P}$-theory and the notion of limit operators, where we focus on Banach space valued sequence spaces ${\mathbf X}$ on general countable discrete groups $({\mathcal G},+)$.

~

{\bf Fredholm operators and essential spectrum.}
We start with a complex Banach space ${\mathbf X}$ and denote the sets of all bounded and all compact operators on ${\mathbf X}$ by ${\mathcal L}({\mathbf X})$ and ${\mathcal K}({\mathbf X})$, respectively. Recall that ${\mathcal K}({\mathbf X})$ forms a closed two-sided ideal in the Banach algebra ${\mathcal L}({\mathbf X})$. An operator $A\in{\mathcal L}({\mathbf X})$ is a {\em Fredholm operator} if its kernel has finite dimension and its image has finite co-dimension. By Calkin's theorem this is equivalent to invertibility of the coset $A+{\mathcal K}({\mathbf X})$ in the quotient algebra ${\mathcal L}({\mathbf X})/{\mathcal K}({\mathbf X})$ -- also known as the Calkin algebra. For an operator $A\in {\mathcal L}({\mathbf X})$ we write $\spec(A)$ for its spectrum and $\specess(A)$ for its {\em essential spectrum}, where the latter refers to the set of all $\lambda\in{\mathbb C}$ such that $A-\lambda I$ is not Fredholm; it is the spectrum of the coset $A+{\mathcal K}({\mathbf X})$ in the Calkin algebra.

~

{\bf Limit operators.}
An important (and in our setting the only) part of the spectrum of an operator is the essential spectrum, leading to the study of Fredholm operators, i.e., invertibility modulo compact operators. The first successful study for a large class of operators, the so-called band-dominated operators, on a space ${\mathbf X}=\ell^p({\mathbb Z}^N,Y)$ with $\dim Y<\infty$ and $p\in (1,\infty)$, was done by Lange and Rabinovich \cite{LangeRabinovich1985} and finally with full proofs in \cite{RabinovichRochSilbermann1998} by Rabinovich, Roch and Silbermann, both are based on works of Muhamadiev \cite{Muh3} and Simonenko \cite{Simonenko68}. See Section 1.2 of \cite{CL10} for a more detailed history of the subject.

The argument in a nutshell: The Fredholm property of an operator $A$ on ${\mathbf X}$ is invariant under compact perturbations such as arbitrary modifications to finitely many entries of the matrix $(a_{ij})_{i,j\in{\mathbb Z}^N}$ that is associated to $A$; so the property must be ``encoded'' in this matrix ``at infinity''. To gather this information, one looks at limits of translates of the matrix (or, likewise, the operator $A$) as translation goes to infinity. Such limits are called limit operators of $A$, and there are usually many of them, each associated to another sequence of translations. \cite{RabinovichRochSilbermann1998} shows that $A$ is Fredholm iff all its limit operators are invertible (and their inverses are uniformly bounded -- this condition was later shown to be redundant \cite{LindnerSeidel2014}), so that the essential spectrum of $A$ is the union of spectra of its limit operators.

~

{\bf ${\mathcal P}$-theory.}
An important aspect is the topology in which limits of translates of $A$ are considered. This topology should be
such that exactly the compact operators disappear (meaning that all their limit operators are zero).

In the simple case ${\mathbf X}=\ell^2({\mathbb Z}^N,{\mathbb C})$, compact operators are easily described:
If we write $P_n$ for the operator on ${\mathbf X}$ of multiplication by the characteristic function
of the set $\{-n,...,n\}^N\subset{\mathbb Z}^N$, then $K\in{\mathcal L}({\mathbf X})$ is compact if and only if both
\begin{equation}\label{eq:comp}
\|K(I-P_n)\|\to 0\qquad\text{and}\qquad\|(I-P_n)K\|\to 0\qquad\text{as}\quad n\to\infty.
\end{equation}
On the matrix level, this means that $K=(k_{ij})$ has finite support or is the norm limit of a sequence
of such matrices.

The characterization of compact operators by \eqref{eq:comp} still holds if ${\mathbf X}=\ell^p({\mathbb Z}^N,Y)$ with $p\in(1,\infty)$ and $\dim Y<\infty$ but
it is no longer true if $p\in\{1,\infty\}$ and/or $\dim Y=\infty$. (See e.g. Example~\ref{ex:RandSchr} below
for a natural instance of ${\mathbf X}=\ell^2({\mathbb Z}^N,Y)$ with $\dim Y=\infty$.)

The ${\mathcal P}$-theory is aiming to unify all these cases by putting the sequence ${\mathcal P}:=(P_n)$ at center stage
and going from there -- with tailor-made versions of compactness, Fredholmness and convergence.
This is not limited to the particular sequence ${\mathcal P}$ introduced above and not even to $\ell^p$-spaces ${\mathbf X}$:

~

Let ${\mathbf X}$ be a complex Banach space. Following the lines of \cite{RabinovichRochSilbermann2004,SeidelSilbermann2012,Seidel2014}, we say that a bounded sequence ${\mathcal P}:=(P_n)_{n\in{\mathbb N}}$ of operators in ${\mathcal L}({\mathbf X})$ is called an \emph{approximate projection} whenever
\begin{itemize}
\item for all $n\in{\mathbb N}$, the operator $P_n$ is not equal to the zero operator respectively the identity operator, and
\item for all $m\in{\mathbb N}$, there exists an $N_m\in{\mathbb N}$ such that $P_nP_m=P_mP_n=P_m$ for all $n\geq N_m$.
\end{itemize}
An approximate projection ${\mathcal P}$ is called an \emph{approximate identity} if for all $x\in{\mathbf X}$ the inequality $\sup_{n\in{\mathbb N}} \|P_nx\|\geq \|x\|$ holds.
This property provides that the elements of the family ${\mathcal P}$ asymptotically exploit the whole space ${\mathbf X}$.
For an approximate projection ${\mathcal P}$, define $S_1:=P_1$ and $S_n:=P_{n+1}-P_n,\; n\geq 1$. Then ${\mathcal P}$ is called \emph{uniform} if the supremum
$\sup\{\|\sum_{k\in U}S_k\|\; :\; U\subseteq{\mathbb N} \text{ finite} \}$ is finite.

\begin{defini}[${\mathcal P}$-compact operators]
  Let ${\mathcal P}:=(P_n)_{n\in{\mathbb N}}$ be an approximate projection. An operator $K\in {\mathcal L}({\mathbf X})$ is called \emph{${\mathcal P}$-compact} if \eqref{eq:comp} holds.
  Denote  by ${\mathcal K}({\mathbf X},{\mathcal P})$ the set of all ${\mathcal P}$-compact operators in ${\mathcal L}({\mathbf X})$.
\end{defini}

One setback is that, unlike ${\mathcal K}({\mathbf X})$, the new set ${\mathcal K}({\mathbf X},{\mathcal P})$ is not an ideal in the algebra ${\mathcal L}({\mathbf X})$ of all bounded linear operators on ${\mathbf X}$. Therefore pass to a smaller algebra -- the so-called multiplier algebra of ${\mathcal K}({\mathbf X},{\mathcal P})$ in ${\mathcal L}({\mathbf X})$. This is the set
\[{\mathcal L}({\mathbf X},{\mathcal P})\ :=\ \{A\in {\mathcal L}({\mathbf X}):\; AK, KA\in {\mathcal K}({\mathbf X},{\mathcal P})\,\text{for all}\, K\in {\mathcal K}({\mathbf X},{\mathcal P})\}.\]
Then ${\mathcal L}({\mathbf X},{\mathcal P})$ is a closed, unital Banach subalgebra of ${\mathcal L}({\mathbf X})$ containing ${\mathcal K}({\mathbf X},{\mathcal P})$ as a closed two-sided ideal, see \cite[Proposition 1.1.8]{RabinovichRochSilbermann2004}.
In Figure~\ref{fig:venn_diagramm} an illustration of the relations between the different algebras is sketched.

\begin{defini}[${\mathcal P}$-Fredholm operators]
We say that $A\in {\mathcal L}({\mathbf X},{\mathcal P})$ is a \emph{${\mathcal P}$-Fredholm operator} if $A$ is invertible in the quotient algebra ${\mathcal L}({\mathbf X},{\mathcal P})/{\mathcal K}({\mathbf X},{\mathcal P})$.
\end{defini}
By \cite[Corollary 12]{Seidel2014}, every Fredholm operator in ${\mathcal L}({\mathbf X},{\mathcal P})$ is a ${\mathcal P}$-Fredholm operator. The reverse implication requires $\dim Y$ to be finite.

\begin{figure}[tb]
  \setlength{\unitlength}{0.78mm}
  \begin{tabular}{|c|c|c|}
  \hline
  \rule{0mm}{2.5ex} & $\dim Y<\infty$ & $\dim Y=\infty$\\
  \hline
  \hline
  \begin{picture}(20,62)
    \put(10,31){\makebox(0,0)[]{$1<p<\infty$}}
  \end{picture} &

  \begin{picture}(60,62)
    \put(30,30){\oval(60,60)[]}
    \put(3,58){\makebox(0,0)[tl]{${\mathcal L}({\mathbf X})={\mathcal L}({\mathbf X},{\mathcal P})$}}
    \put(38,22){\oval(40,40)[]}
    \put(38,40){\makebox(0,0)[t]{${\mathcal K}({\mathbf X})={\mathcal K}({\mathbf X},{\mathcal P})$}}
    \put(9.5,22){\circle{0.5}}
    \put(10,22.5){\makebox(0,0)[bl]{\scriptsize $I$}}
  \end{picture} &

  \begin{picture}(60,62)
    \put(30,30){\oval(60,60)[]}
    \put(3,58){\makebox(0,0)[tl]{${\mathcal L}({\mathbf X})$}}
    \put(34,26){\oval(48,48)[]}
    \put(13,48){\makebox(0,0)[tl]{${\mathcal L}({\mathbf X},{\mathcal P})$}}
    \put(38,22){\oval(36,36)[]}
    \put(23,38){\makebox(0,0)[tl]{${\mathcal K}({\mathbf X},{\mathcal P})$}}
    \put(42,18){\oval(24,24)[]}
    \put(33,28){\makebox(0,0)[tl]{${\mathcal K}({\mathbf X})$}}
    \put(14.5,22){\circle{0.5}}
    \put(15,22.5){\makebox(0,0)[bl]{\scriptsize $I$}}
  \end{picture} \\

  \hline
  \begin{picture}(20,62)
    \put(10,31){\makebox(0,0)[]{$p=1,\infty$}}
  \end{picture} &

  \begin{picture}(60,62)
    \put(30,30){\oval(60,60)[]}
    \put(3,58){\makebox(0,0)[tl]{${\mathcal L}({\mathbf X})$}}
    \put(26,33){\oval(40,40)[]}
    \put(9,51){\makebox(0,0)[tl]{${\mathcal L}({\mathbf X},{\mathcal P})$}}
    \put(40,20){\oval(35,35)[]}
    \put(54,4){\makebox(0,0)[br]{${\mathcal K}({\mathbf X})$}}
    \put(34,25){\makebox(0,0)[]{${\mathcal K}({\mathbf X},{\mathcal P})$}}
    \put(14.5,30){\circle{0.5}}
    \put(15,30.5){\makebox(0,0)[bl]{\scriptsize $I$}}
  \end{picture} &

  \begin{picture}(60,62)
    \put(30,30){\oval(60,60)[]}
    \put(3,58){\makebox(0,0)[tl]{${\mathcal L}({\mathbf X})$}}
    \put(26,33){\oval(40,40)[]}
    \put(9,51){\makebox(0,0)[tl]{${\mathcal L}({\mathbf X},{\mathcal P})$}}
    \put(42,18){\oval(31,31)[]}
    \put(54,4){\makebox(0,0)[br]{${\mathcal K}({\mathbf X})$}}
    \put(31,28){\oval(30,30)[]}
    \put(19,41){\makebox(0,0)[tl]{${\mathcal K}({\mathbf X},{\mathcal P})$}}
    \put(11,30){\circle{0.5}}
    \put(11.5,30.5){\makebox(0,0)[bl]{\scriptsize $I$}}
  \end{picture} \\
  \hline
  \end{tabular}
\caption{The spaces ${\mathcal L}({\mathbf X})$, ${\mathcal L}({\mathbf X},{\mathcal P})$, ${\mathcal K}({\mathbf X},{\mathcal P})$ and ${\mathcal K}({\mathbf X})$ (the compact operators on ${\mathbf X}$). See\ \cite{Lindner2006,Seidel2014}}
  \label{fig:venn_diagramm}
\end{figure}

Compact operators are exactly the operators which, when multiplied from the right, turn strong
convergence into norm-convergence. This property is mimicked now in the ${\mathcal P}$-setting:
\begin{defini}[${\mathcal P}$-strong convergence]
  Let ${\mathcal P}:=(P_n)_{n\in{\mathbb N}}$ be an approximate identity. A sequence $(A_n)$ in ${\mathcal L}({\mathbf X})$ \emph{converges ${\mathcal P}$-strongly} to $A\in {\mathcal L}({\mathbf X})$ if
  \[\lim\limits_{n\to\infty}\norm{K(A_n-A)} + \norm{(A_n-A)K} = 0\]
  for all $K\in{\mathcal K}({\mathbf X},{\mathcal P})$. In this case, we write $A = \Plim A_n$ or $A_n\pto A$.
\end{defini}

${\mathcal L}({\mathbf X},{\mathcal P})$ is the natural playground for the study of ${\mathcal P}$-strong convergence. For example, ${\mathcal P}$-strong convergence of a sequence $(A_n)$ from ${\mathcal L}({\mathbf X},{\mathcal P})$ implies that the sequence is bounded, the limit is unique and again in ${\mathcal L}({\mathbf X},{\mathcal P})$, see \cite[Section 1.1.4]{RabinovichRochSilbermann2004}.

For a sequence $(A_n)_{n\in{\mathbb N}}$ in ${\mathcal L}({\mathbf X},{\mathcal P})$ we have $A = \Plim A_n$ if and only if $(A_n)$ is bounded and $$\lim\limits_{n\to\infty}\norm{P_m(A-A_n)} + \norm{(A-A_n)P_m}= 0$$
for all $m\in{\mathbb N}$, see \cite[Proposition 1.1.4]{RabinovichRochSilbermann2004}.

\begin{rem}
\label{rem:P-topology}
  Let ${\mathcal P}:=(P_n)_{n\in{\mathbb N}}$ be an approximate identity. For $n\in{\mathbb N}$ define the seminorm $p_n: {\mathcal L}({\mathbf X},{\mathcal P})\to {\mathbb R}$ by
  $$p_n(A) := \norm{P_n A} + \norm{AP_n},  \qquad A\in {\mathcal L}({\mathbf X},{\mathcal P}).$$
  Then, on bounded subsets of ${\mathcal L}({\mathbf X},{\mathcal P})$, the family $(p_n)_{n\in{\mathbb N}}$ of seminorms defines a metriziable topology, for which the ${\mathcal P}$-limit is the limit with respect to this metric.
\end{rem}

\begin{rem}
Let ${\mathbf X}=\ell^p({\mathbb Z}^N,Y)$ with $1<p<\infty$ and $\dim Y <\infty$, and let ${\mathcal P}=(P_n)$, where $P_n$ is
the operator of multiplication by the characteristic function of $\{-n,...,n\}^N\subset{\mathbb Z}^N$.
Then ${\mathcal L}({\mathbf X},{\mathcal P}) = {\mathcal L}({\mathbf X})$, ${\mathcal P}$-strong convergence is equivalent to $*$-strong convergence (that is strong convergence $A_n\to A$ and $A_n^*\to A^*$), and ${\mathcal P}$-Fredholmness coincides with Fredholmness.
\end{rem}

Historically, ${\mathcal P}$-theory has its roots in Simonenko \cite{Simonenko68}, started in the present form with
Pr\"ossdorf, Roch and Silbermann \cite{ProssdorfSilbermann,RochSilbermann1989} and recently culminated
in Seidels work -- see \cite{Seidel2014} for his excellent survey.

~

{\bf Operators on discrete groups.}
Now we pass to $\ell^p$-spaces on a discrete group ${\mathcal G}$ rather than fixing ${\mathcal G}={\mathbb Z}^N$.
So let $p\in[1,\infty]$, $Y$ a Banach space, $({\mathcal G},+)$ a discrete group (not nessesarily abelian or finitely generated), and let ${\mathbf X}:=\ell^p({\mathcal G},Y)$.

For $n\in{\mathbb N}$ let ${\mathcal G}_n\subseteq {\mathcal G}$ be such that $\varnothing\neq {\mathcal G}_n \neq {\mathcal G}$, and such that for all $m\in{\mathbb N}$ there exists $N_m$ satisfying ${\mathcal G}_m\subseteq {\mathcal G}_n$ for all $n\geq N_m$. Furthermore, suppose $\bigcup_{n\in{\mathbb N}} {\mathcal G}_n = {\mathcal G}$. For $x\in{\mathbf X}$ define
$$P_n(x_g)_{g\in {\mathcal G}} := ({\mathbf 1}_{{\mathcal G}_n}(g) x_g)_{g\in {\mathcal G}}$$
where ${\mathbf 1}_{{\mathcal G}_n}$ denotes the characteristic function of the set ${\mathcal G}_n$. Then ${\mathcal P}:=(P_n)_{n\in{\mathbb N}}$ defines an approximate identity which can be checked by inspection.
Furthermore, ${\mathcal P}$ is even uniform.
In the following we always suppose that ${\mathcal P}$ is defined by such an exhaustion defining a uniform approximate identity.

For $g\in {\mathcal G}$ and $x\in {\mathbf X}$ let $V_gx:=(x_{h+g})_{h\in {\mathcal G}}$ denote the \emph{shift by $g$}. For a sequence $(g_n)_{n\in{\mathbb N}}$ in ${\mathcal G}$ we write $g_n \to \infty$ if it leaves every finite set, i.e.,\ if for every finite set $F\subseteq {\mathcal G}$ there exists an $n_F\in{\mathbb N}$ with $g_n\notin F$ for all $n\geq n_F$.
With this notation at hand we define limit operators:

\begin{defini}[limit operators]
Let $A\in {\mathcal L}({\mathbf X},{\mathcal P})$ and $g:=(g_n)_{n\in{\mathbb N}}$ in ${\mathcal G}$ with $g_n\to \infty$. Then
\[A_g\ :=\ \Plim_{n\to\infty} V_{g_n}AV_{-g_n},\]
if it exists, is called the \emph{limit operator of $A$ with respect to $g$}.
The set $\sop(A)$ of all limit operators of $A$ is called the \emph{operator spectrum of $A$}. An operator $A\in {\mathcal L}({\mathbf X},{\mathcal P})$ is called \emph{self-similar} if $A\in \sop(A)$.
\end{defini}

The importance of the operator spectrum becomes clear in the study of Fredholm and ${\mathcal P}$-Fredholm operators. For so-called rich band-dominated operators on ${\mathbf X}$, which is a proper subalgebra of ${\mathcal L}({\mathbf X},{\mathcal P})$, we have, by \cite{Seidel2014,LindnerSeidel2014}, that
\[
A \textrm{ is Fredholm}\quad\Longrightarrow\quad
A \textrm{ is ${\mathcal P}-$Fredholm}\quad\iff\quad
\textrm{ all } B\in\sop(A) \textrm{ are invertible.}
\]
Both ``$\Longrightarrow$'' arrows already hold in ${\mathcal L}({\mathbf X},{\mathcal P})$, so that we have the following theorem:

\begin{theo}[see {\cite[Theorem 1.28]{SeidelSilbermann2012}}]
\label{thm:spectral_inclusion_groups} Let
$A \in {\mathcal L}({\mathbf X},{\mathcal P})$. Then
$$
\specess(A)\ \supseteq\ \underset{B\in\sop(A)}{\bigcup}\spec(B)
$$
holds. In particular,
$$
\spec(A)=\specess(A)
$$
holds if $A$ is self-similar.
\end{theo}

\section{Dynamical systems over discrete groups}
\label{sec:dynamical_systems}
\bigskip

Let $({\mathcal G},+)$ be a discrete group with countably many elements,
$\Omega$ a compact metric space, such that there exists a
representation $\alpha$ of ${\mathcal G}$ in $\{T:\Omega\to\Omega:\;
T\,\text{homeomorphism}\}$, i.e., $\alpha\colon {\mathcal G}\to \{T:\Omega\to\Omega:\;
T\,\text{homeomorphism}\}$ such that
\[\alpha(g+h) = \alpha(g)\circ\alpha(h) \quad(g,h\in {\mathcal G}).\]
Then we call $(\Omega,{\mathcal G},\alpha)$ a \emph{dynamical system}.
Note that if ${\mathcal G}$ is abelian, then
the homeomorphisms in the image of $\alpha$ commute.

\begin{defini}[limit sets]
Let $(\Omega,{\mathcal G}, \alpha)$ be a dynamical system.
For $\omega\in\Omega$ we write
\[L(\omega):=\{\nu\in\Omega:\; \exists\, (g_n)_{n\in{\mathbb N}} \text{ in  } {\mathcal G} \text{ s.t. }  g_n\to \infty \text{ and } \alpha(g_n)(\omega)\to \nu\},\]
the \emph{limit set of $\omega$}.
\end{defini}

\begin{lemma}\label{lem:properties-L-omega}
Let $(\Omega,{\mathcal G}, \alpha)$ be a dynamical system, $\omega\in\Omega$. Then $L(\omega)$ is compact, non-empty and
$\alpha$-invariant.
\end{lemma}

\begin{proof} For a finite set $F\subseteq {\mathcal G}$ we define
$$L_F (\omega):=\overline{\{ \alpha
(g) (\omega) : g\notin F\}}.$$
Then $L_F (\omega)$ is closed, and
hence, compact. Moreover, as ${\mathcal G}$ is infinite every finite
intersection of sets of the form $L_F (\omega)$, $F\subseteq {\mathcal G}$, is
non-empty. Thus, by compactness, the infinite intersection
$$L(\omega) =\bigcap_{F \subseteq {\mathcal G}, F \,\text{finite}} L_F (\omega)$$
of compact sets is compact and non-empty. Moreover, for each $g\in {\mathcal G}$ the equation
$$\bigcap_{F \subseteq {\mathcal G}, F \,\text{finite}} L_F (\omega) = \bigcap_{F \subseteq {\mathcal G}, F \,\text{finite}} L_{F+g}
(\omega)$$ holds and so the $\alpha$-invariance of $L(\omega)$ follows.
\end{proof}

\begin{defini}[Pseudoergodic elements] Let $(\Omega,{\mathcal G}, \alpha)$ be a dynamical system.
Then  $\omega\in\Omega$  is called \textit{pseudoergodic} if
$L(\omega) = \Omega$ holds. The set of pseudoergodic elements is denoted by $\Omega_\PE$.
\end{defini}

\begin{defini}[Minimality] Let $(\Omega,{\mathcal G}, \alpha)$ be a dynamical system.
For $\omega\in\Omega$ we define the \emph{orbit} of $\omega$ by
\[\Orb(\omega):=\{\alpha(g)(\omega):\; g\in{\mathcal G}\}.\]
We say that $(\Omega,{\mathcal G},\alpha)$ is \emph{minimal} if $\Orb(\omega)$ is dense in $\Omega$ for every $\omega\in\Omega$.
\end{defini}

Clearly, pseudoergodic elements have dense orbits in $\Omega$. We next show the converse under an additional assumption.

\begin{lemma}
\label{lem:pseudo-ergodic}
Let $(\Omega,{\mathcal G}, \alpha)$ be a dynamical system, $\omega\in\Omega$ not isolated and $\Orb(\omega)$ dense in $\Omega$.
Then $\omega$ is pseudoergodic.
\end{lemma}

\begin{proof}
As $\Orb(\omega)$ is dense and $L(\omega)$ is closed and $\alpha$-invariant it suffices to show that it contains
$\omega$. Since $\omega$ is not isolated, there exists a sequence $(\omega_n)$ in $\Omega$
converging to $\omega$ such that the $\omega_n$ are pairwise
different and none of them equals $\omega$. Let $d$ denote a metric on $\Omega$ generating the topology.
By the denseness of the orbit $\Orb(\omega)$, for each $n\in{\mathbb N}$ there exists $g_n\in {\mathcal G}$ with $ d(\alpha(g_n)(\omega), \omega_n) \leq \frac{1}{3} d(\omega, \omega_n)$. The assumption
on the $(\omega_n)$ provides that the set $\{g_n : n\in {\mathbb N}\}$ is
infinite. Thus, $(g_n)_{n\in{\mathbb N}}$ leaves every finite set and the equations
$$\lim_{n\to \infty} \alpha(g_{n})(\omega) = \lim_{n\to
\infty} \omega_{n}  = \omega$$
hold, proving $\omega\in L(\omega)$.
\end{proof}

\begin{proposi}
\label{prop:minimal_all_elements_pseudo}
  Let $(\Omega,{\mathcal G},\alpha)$ be a dynamical system.
  The following assertions are equivalent:
  \begin{itemize}
   \item[$(i)$] $(\Omega,{\mathcal G},\alpha)$ is minimal.
   \item[$(ii)$] $\Omega = \Omega_{\PE}$.
   \item[$(iii)$] $\Omega_{\PE}$ is closed and non-empty.
  \end{itemize}
\end{proposi}

\begin{proof}
  To prove the implication $(i)\Rightarrow (ii)$ we choose $\omega \in \Omega$ and show $L(\omega) = \Omega$.
  By Lemma~\ref{lem:properties-L-omega}, $L(\omega)$ is non-empty, closed and
  $\alpha$-invariant. Hence, it contains at least the closure of one orbit.
  Thus, by minimality it equals $\Omega$. The implication $(ii)\Rightarrow (iii)$ is clear. Finally, $(iii)\Rightarrow (i)$ can be proven as follows. Let $\omega\in\Omega_\PE$. Since $\Omega_\PE$ is $\alpha$-invariant we observe $\Orb(\omega)\subseteq \Omega_\PE$. Since $\Orb(\omega)$ is dense and $\Omega_\PE$ is closed we obtain
  $\Omega = \Omega_\PE$, so every element is pseudoergodic and therefore has a dense orbit.
\end{proof}

\section{Operators over dynamical systems}
\label{sec:operator_over_DS}
\bigskip

In this section we define operators parametrized by dynamical systems and study their spectral theory.

\begin{defini}
\label{def:family_of_operators}
Let $(\Omega,{\mathcal G},\alpha)$ be a dynamical system.
Let $A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ such that
\begin{itemize}
\item[$(i)$] $A(\alpha(g)(\omega))=V_{g}A(\omega)V_{-g}$ for all $\omega\in\Omega$, $g\in {\mathcal G}$.\hfill\textit{(Equivariance)}
\item[$(ii)$] The map $\omega\mapsto A(\omega)$ is ${\mathcal P}$-continuous, i.e.,\ $\omega_n\to \omega$ implies \hfill\textit{(Continuity)}
$$\Plim A(\omega_n)= A(\omega).$$
\end{itemize}
Then $A$ is called a \emph{family of operators over $(\Omega,{\mathcal G},\alpha)$}.
\end{defini}

Let $(\Omega,{\mathcal G},\alpha)$ be a dynamical system and $A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ be a family of operators over $(\Omega,{\mathcal G},\alpha)$.
If $A(\Omega)$ is bounded then the continuity of $A(\cdot)$ as in $(ii)$ is exactly the continuity with respect to the topology as in Remark \ref{rem:P-topology}.

\begin{proposi}
\label{prop:operator_spectra}
Let $(\Omega,{\mathcal G},\alpha)$ be a dynamical system,
$A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ a family of
operators over $(\Omega,{\mathcal G},\alpha)$ and $\omega\in\Omega$. Then
$$
\sop(A(\omega))=\left\{ A(\nu)\;:\; \nu\in L(\omega) \right\}
$$
for all $\omega\in\Omega$.
\end{proposi}

\begin{proof}
We first show ``$\subseteq$''. Let $\omega\in \Omega$ and $B\in\sop(A(\omega))$.
Then there exists a sequence $(g_n)_{n\in{\mathbb N}}$ in ${\mathcal G}$
with $g_n\to\infty$, such that
$$
B=\Plim
V_{g_n} A(\omega) V_{-g_n} =\Plim A(\alpha(g_n)(\omega)).
$$
Since $g_n\to\infty$ and $\Omega$ is compact we can select a subsequence
$(g_{n_k})_{k\in{\mathbb N}}$ such that $(\alpha(g_{n_k})(\omega))_{k\in{\mathbb N}}$ is
convergent to $\nu\in \Omega$.
Thus, $\nu \in L(\omega)$. Using
the ${\mathcal P}$-continuity of $A(\cdot)$ we obtain
$B = A(\nu)$.

\medskip

We now turn to prove the converse inclusion ``$\supseteq$''. For $\omega\in\Omega$ and $\nu \in L(\omega)$,
there is a sequence $(g_n)_{n\in{\mathbb N}}$ such that $g_n\to\infty$ and $\lim_{n\to\infty} \alpha(g_n)(\omega)=\nu$.
Then the ${\mathcal P}$-continuity of $A(\cdot)$
implies
$$
A(\nu)=\Plim
A(\alpha(g_n)(\omega))= \Plim V_{g_n}A(\omega)V_{-g_n},
$$
leading to $A(\nu)\in\sop(A(\omega))$.
\end{proof}

\begin{coro}
\label{cor:operator_spectrum_pseudoergodic}
Let $(\Omega,{\mathcal G}, \alpha)$ be a dynamical system and $A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ a family of operators over $(\Omega,{\mathcal G},\alpha)$. Then
$$
\sop(A(\omega)) = \left\{ A(\nu)\;:\; \nu\in \Omega \right\}
$$
for all $\omega\in \Omega_\PE$. In particular, $A(\omega)$ is self-similar for each $\omega\in \Omega_\PE$.
\end{coro}

\begin{proof}
Since $\omega$ is pseudoergodic we have $L(\omega) = \Omega$. Thus, the assertion follows from the previous Proposition~\ref{prop:operator_spectra}.
\end{proof}

\begin{theo}
\label{thm:constant_spectra_pseudo-ergodic_groups}
Let $(\Omega,{\mathcal G}, \alpha)$ be a dynamical system and $A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ a family of operators over $(\Omega,{\mathcal G},\alpha)$.
Set
$$
\Sigma:=\underset{\omega\in \Omega}{\bigcup}\spec(A(\omega)).
$$
Then
$$
\spec(A(\omega))=\specess(A(\omega))=\Sigma
$$
for all $\omega\in \Omega_{\PE}$.
\end{theo}

\begin{proof}
  For pseudoergodic $\omega\in\Omega$ we have, taking into account Theorem \ref{thm:spectral_inclusion_groups} and Corollary \ref{cor:operator_spectrum_pseudoergodic},
  \[\Sigma\supseteq \spec(A(\omega)) \supseteq \specess(A(\omega))\supseteq\underset{B\in\sop(A)}{\bigcup}\spec(B)  \supseteq\underset{\nu\in\Omega}{\bigcup}\spec(A(\nu)) = \Sigma. \qedhere\]
\end{proof}

\begin{coro}
\label{cor:constant_spectra_minimal_groups}
Let $(\Omega,{\mathcal G},\alpha)$ be a minimal dynamical system and $A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ a family of operators over $(\Omega,{\mathcal G},\alpha)$.
Set
$$
\Sigma:=\underset{\omega\in\Omega}{\bigcup}\spec(A(\omega)).
$$
Then
$$
\spec(A(\omega))=\specess(A(\omega))=\Sigma
$$
for all $\omega\in \Omega$.
\end{coro}

\begin{proof}
According to Proposition~\ref{prop:minimal_all_elements_pseudo} minimality implies $\Omega_{\PE} = \Omega$. Thus, the statement
follows directly from the previous Theorem~\ref{thm:constant_spectra_pseudo-ergodic_groups}.
\end{proof}

As we have seen, the spectrum of a family of operators over a minimal dynamical system is constant.
Next, we show that in this setup also the norms of the resolvents are equal for all operators of the family.
While this is an immediate consequence of spectral constancy in case of selfadjoint operators, in the non-selfadjoint case this might be a bit surprising.
We will need a little preparation.

\begin{lemma}
\label{lem:limit_operator_norm}
  Let $A\in {\mathcal L}({\mathbf X},{\mathcal P})$, $B\in \sop(A)$. Then $\|B\|\leq \|A\|$.
\end{lemma}

\begin{proof}
  This is a direct consequence of \cite[Proposition 1.1.17]{RabinovichRochSilbermann2004} and the fact that $\|P_n\|\leq 1$ for all $n\in{\mathbb N}$.
\end{proof}

\begin{theo}
\label{thm:resovlent_norms}
  Let $(\Omega,{\mathcal G},\alpha)$ be a dynamical system, $A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ a family of operators over $(\Omega,{\mathcal G},\alpha)$.
  Let $\Sigma:=\underset{\omega\in\Omega}{\bigcup}\spec(A(\omega))$,
  $\lambda\in{\mathbb C}\setminus \Sigma$, and $\omega,\nu\in\Omega_\PE$. Then
  \[\|(\lambda-A(\omega))^{-1}\| = \|(\lambda-A(\nu))^{-1}\|.\]
  If $(\Omega,{\mathcal G},\alpha)$ is minimal then
  \[\|(\lambda-A(\omega))^{-1}\| = \|(\lambda-A(\nu))^{-1}\|\]
  holds for all $\omega,\nu\in\Omega$.
\end{theo}

\begin{proof}
  Clearly, $\sop(\lambda-A(\omega)) = \lambda-\sop(A(\omega))$. Therefore, without loss of generality, we may suppose that $\lambda=0$.
  Since $A(\omega)^{-1}$ trivially is a regularizer for $A(\omega)$, i.e.,\ $A(\omega)^{-1}A(\omega)-I, A(\omega)A(\omega)^{-1}-I\in {\mathcal K}({\mathbf X},{\mathcal P})$,
  by \cite[Theorem 16]{Seidel2014} (which was proven there only for ${\mathcal G}={\mathbb Z}^N$,
  but generalizes to arbitrary groups without any difficulty) we have
  \[\sop(A(\omega)^{-1}) = \{B^{-1}:\; B\in\sop(A(\omega))\} = \{A(\nu)^{-1}:\; \nu\in\Omega\}\]
  due to Corollary \ref{cor:operator_spectrum_pseudoergodic}.
  Thus, by Lemma \ref{lem:limit_operator_norm} we obtain
  \[\|A(\nu)^{-1}\| \leq  \|A(\omega)^{-1}\|.\]
  Interchanging the roles of $\omega$ and $\nu$ yields the first part of our assertion.

  In case of minimality, we just note that $\Omega_\PE = \Omega$ holds by Proposition \ref{prop:minimal_all_elements_pseudo}.
\end{proof}

The super-level sets of the resolvent norm yield the so-called pseudospectra:

\begin{defini}[pseudospectrum]
  Let $A\in {\mathcal L}({\mathbf X})$, $\varepsilon>0$. Then we define the \emph{$\varepsilon$-pseudospectrum of $A$} by
  \[\spec_\varepsilon(A):=\{\lambda\in{\mathbb C}:\; \|(\lambda-A)^{-1}\|> \tfrac{1}{\varepsilon}\},\]
  where we set $\|(\lambda-A)^{-1}\|:=\infty$ if $\lambda-A$ is not boundedly invertible.
\end{defini}

For normal (in particular selfadjoint) operators $A$, $\spec_\varepsilon(A)$ is the $\varepsilon$-neighbourhood of the spectrum.
In general, it is larger. Part of the attraction of pseudospectra is that $\spec_\varepsilon(A)$ equals the union of
$\spec(A+T)$ over all perturbations $\|T\|<\varepsilon$. The interplay between resolvents of $A$ and semigroup $(e^{tA})_{t\geq 0}$ moreover
allows bounds on the transient growth of the semigroup in terms of the pseudospectrum (whereas the spectrum just determines
the asymptotics as $t\to\infty$). See the monograph \cite{TrefethenEmbree2005} for more aspects of pseudospectra.

In view of Theorem \ref{thm:resovlent_norms} we can also obtain equality of pseudospectra for families of operators.

\begin{coro}
\label{cor:pseudospectra}
  Let $(\Omega,{\mathcal G},\alpha)$ be a dynamical system, $A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ a family of operators over $(\Omega,{\mathcal G},\alpha)$, $\varepsilon>0$.
  Then
  \[\spec_\varepsilon(A(\omega)) = \spec_\varepsilon(A(\nu))\]
  holds for all $\omega,\nu\in\Omega_\PE$.
  If $(\Omega,{\mathcal G},\alpha)$ is minimal then
  \[\spec_\varepsilon(A(\omega)) = \spec_\varepsilon(A(\nu))\]
  holds for all $\omega,\nu\in\Omega$.
\end{coro}

Note that Corollary \ref{cor:pseudospectra} contains our corresponding theorem on spectra as a special case (after letting $\varepsilon\to 0$).

\begin{exam}[Operators of finite propagation]
  Let $A\in {\mathcal L}({\mathbf X})$. Then we say that $A$ is \emph{of finite propagation} if
  \[\prop(A):=\sup\{d(g,h):\; g,h\in{\mathcal G},\, (A\delta_g)(h) = 0\}<\infty,\]
  where $d$ is the path distance in ${\mathcal G}$ and
  \[\delta_g(h):=\begin{cases} 1 & h = g,\\ 0 & \text{else}\end{cases} \quad(g\in{\mathcal G}).\]
  Then, $\prop(A)$ is called \emph{propagation} of $A$. Note that each operator of finite propagation belongs to ${\mathcal L}({\mathbf X},{\mathcal P})$.

  Now, let $A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ be a family of operators over $(\Omega,{\mathcal G},\alpha)$ with finite propagation.
  Typical models of this kind appear in the study of Schr\"odinger operators with random potentials on ${\mathcal G}$.
\end{exam}

\begin{exam}[Band-dominated operators]
  Since ${\mathcal L}({\mathbf X},{\mathcal P})$ is closed in ${\mathcal L}({\mathbf X})$, norm-limits of operators of finite propagation are contained in ${\mathcal L}({\mathbf X},{\mathcal P})$.
  Such operators are called \emph{band-dominated operators}, and are typically studied for the case ${\mathcal G}={\mathbb Z}^N$, see e.g.~\cite{RabinovichRochSilbermann1998,RabinovichRochSilbermann2004,Lindner2006,CL10,Seidel2014},
  and on $\ell^2({\mathcal G})$ for a finitely generated group ${\mathcal G}$ in \cite{Roe2005} by means of $C^*$-algebra techniques.
\end{exam}

\section{Operators over finitely generated groups}
\label{sec:operators_over_finitely_generated_groups}

\bigskip

Let us now assume that $({\mathcal G},+)$ is finitely generated, i.e.,\ there exists ${\mathcal S}\subseteq {\mathcal G}$ finite such that for every $g\in{\mathcal G}$ there exist $n\in{\mathbb N}$ and $s_1,\ldots,s_n\in{\mathcal S}\cup(-{\mathcal S})$ such that $g = s_1+\ldots+s_n$.

Let $\Omega$ be a compact metric space and for $s\in{\mathcal S}$ let $\alpha(s):\Omega\to\Omega$ be a homeomorphism. We may assume that
$\alpha(-s) = \alpha(s)^{-1}$ for all $s\in {\mathcal S}$ (either it is a definition of $\alpha(-s)$ or it is an assumption, depending on whether $-s\in{\mathcal S}$ or not), and that
if $g = s_1+s_2+\ldots s_n\in{\mathcal G}$ then
\[\alpha(g):=\alpha(s_1)\circ \alpha(s_2)\circ\ldots\circ \alpha(s_n)\]
is well-defined, i.e.,\ if $g = s_1+\ldots+s_n = s_1'+\ldots +s_m'$ then
\[\alpha(s_1)\circ\ldots\circ \alpha(s_n) = \alpha(s_1')\circ\ldots\circ \alpha(s_m').\]

\begin{lemma}
  For $g,h\in{\mathcal G}$ we have
  $\alpha(g+h) = \alpha(g)\circ\alpha(h)$. Hence, $(\Omega,{\mathcal G},\alpha)$ is a dynamical system.
\end{lemma}

\begin{proof}
  Write $g = s_1+\ldots+s_n$ and $h = s_1'+\ldots+s_m'$ with $s_1,\ldots,s_n,s_1',\ldots,s_m'\in{\mathcal S}\cup(-{\mathcal S})$. Then $g+h = s_1+\ldots+s_n + s_1'+\ldots+s_m'$ and so the equalities
  \begin{align*}
    \alpha(g+h) & = \alpha(s_1)\circ\ldots \circ\alpha(s_n)\circ\alpha(s_1')\circ\ldots\circ\alpha(s_m') = \alpha(g)\circ\alpha(h)
  \end{align*}
  hold.
\end{proof}

Recall that for $g\in{\mathcal G}$ we have defined the shift $V_gx:=(x_{h+g})_{h\in{\mathcal G}}$ $(x\in {\mathbf X})$.
Consider $A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ such that
\begin{itemize}
\item[$(i')$] $A(\alpha(s)(\omega))=V_{s}A(\omega)V_{-s}$ for all $\omega\in\Omega$, $s\in {\mathcal S}$.\hfill\textit{(Equivariance)}
\item[$(ii)$] The map $\omega\mapsto A(\omega)$ is ${\mathcal P}$-continuous.\hfill\textit{(Continuity)}
\end{itemize}

\begin{lemma}
\label{lem:suff_cond_family_operator}
  Let $A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ satisfy condition $(i')$. Then, for $g\in{\mathcal G}$, $\omega\in\Omega$, we have
  \[A(\alpha(g)(\omega)) = V_{g} A(\omega)V_{-g},\]
  i.e.,\ $A$ fulfills $(i)$ in Definition \ref{def:family_of_operators}. If in addition $A$ also satisfies $(ii)$, then, $A$ is a family of operators over $(\Omega,{\mathcal G},\alpha)$.
\end{lemma}

\begin{proof}
  Write $g = s_1+\ldots+s_n$ with $s_1,\ldots,s_n\in{\mathcal S}\cup(-{\mathcal S})$.
  This yields
  \begin{align*}
    A(\alpha(g)(\omega)) & = A(\alpha(s_1)\circ \alpha(s_2)\circ\ldots\circ \alpha(s_n)(\omega)) = V_{s_1} A(\alpha(s_2)\circ\ldots\circ \alpha(s_n)(\omega))V_{-s_1} \\
    & = \ldots = V_{s_1}\ldots V_{s_n} A(\omega) V_{-s_n} \ldots V_{-s_1} = V_{g} A(\omega) V_{-g}. \qedhere
  \end{align*}
\end{proof}

Thus, we can apply Corollary \ref{cor:constant_spectra_minimal_groups} to obtain constancy of the spectrum whenever $(\Omega,{\mathcal G},\alpha)$ is minimal.

\begin{exam}[Operators on the discrete Heisenberg group]
  We now give an example of a non-abelian group, where our results can be applied as well. Let ${\mathcal G}:=H_3({\mathbb Z})$ be the discrete Heisenberg group, i.e.\ the group of $3\times 3$-matrices of the form
  \[\begin{pmatrix}
      1 & a & c \\
      0 & 1 & b \\
      0 & 0 & 1
    \end{pmatrix}\]
  with $a,b,c\in{\mathbb Z}$, equipped with matrix multiplication. The group is non-abelian and generated by the two elements described by $(a,b,c)\in \set{(1,0,0),(0,1,0)}$.
  Typical examples of operators on the discrete Heisenberg group are Schr\"odinger operators, which are operators of finite propagation.
  They can be used to model random walks on the corresponding Cayley graph of the group, see for example \cite{BeguinValetteZuk1997}.
\end{exam}

\section{Operators over ${\mathbb Z}^N$}
\label{sec:operators_over_ZN}
\bigskip

A particular example of a finitely generated group ${\mathcal G}$ is ${\mathbb Z}^N$, with generators $e_j:=(\delta_{jk})_{k\in\{1,\ldots,N\}}$ ($j\in\{1,\ldots,N\}$), where $\delta$ denotes the Kronecker delta.
Let $\Omega$ be a compact metric space. For $j\in\{1,\ldots,N\}$ let $\alpha(e_j):\Omega\to\Omega$ be a homeomorphism. Assume that $\alpha(e_j)\circ\alpha(e_k) = \alpha(e_k) \circ \alpha(e_j)$ holds for all $j,k\in\{1,\ldots,N\}$.
We now extend $\alpha$ to ${\mathbb Z}^N$: for $g\in{\mathbb Z}^N$ define
\[\alpha(g):=\alpha(e_1)^{g_1}\circ \ldots\circ \alpha(e_n)^{g_n}.\]
The fact that $\alpha(e_j)$ commutes with $\alpha(e_k)$ for all $j,k\in \{1,\ldots,n\}$ yields that $\alpha(g)$ is well-defined. By definition, $\alpha$ is a representation of ${\mathbb Z}^N$ in the set of homeomorphisms on $\Omega$.
Thus, $(\Omega,{\mathbb Z}^N,\alpha)$ is a dynamical system.

Consider $A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ such that
\begin{itemize}
\item[$(i')$] $A(\alpha(e_j)(\omega))=V_{e_j}A(\omega)V_{-e_j}$ for all $\omega\in\Omega$, $j\in\{1,\ldots,N\}$.\hfill\textit{(Equivariance)}
\item[$(ii)$] The map $\omega\mapsto A(\omega)$ is ${\mathcal P}$-continuous.\hfill\textit{(Continuity)}
\end{itemize}
According to Lemma~\ref{lem:suff_cond_family_operator}, the map $A:\Omega\to {\mathcal L}({\mathbf X},{\mathcal P})$ defines a family of operators over $(\Omega,{\mathbb Z}^N,\alpha)$.
Thus, if $(\Omega,{\mathbb Z}^N,\alpha)$ is minimal, then, Corollary \ref{cor:constant_spectra_minimal_groups} yields constancy of the spectrum.

\medskip

Let us remark that much more is known for operator families over ${\mathbb Z}^N$.
Let $S^{N-1}:=\{\eta \in {\mathbb R}^N:\; \abs{\eta} = 1\}$ be the unit sphere in ${\mathbb R}^N$. For $R>0$, $\eta\in S^{N-1}$ and a neighborhood $U\subseteq S^{N-1}$ of $\eta$ we call
\[W_{R,U} := \{k\in {\mathbb Z}^N:\; \abs{k}>R,\, \tfrac{k}{\abs{k}}\in U\}\]
a \emph{neighborhood at infinity of $\eta$}.
Let $g = (g_n)_{n\in{\mathbb N}}$ be a sequence in ${\mathbb Z}^N$ such that $g_n\to \infty$. We say that \emph{$g$ tends to infinity in the direction of $\eta\in S^{N-1}$} if for every neighborhood at infinity $W_{R,U}$ of $\eta$, there exists $n_0\in{\mathbb N}$ such that
\[h_n\in W_{R,U} \quad(n\geq n_0).\]
Let $\eta\in S^{N-1}$ and $A\in {\mathcal L}({\mathbf X},{\mathcal P})$. The \emph{local operator spectrum} $\sop_\eta(A)$ of $A$ at $\eta$ is the set of all limit operators $A_g$ of $A$ with respect to sequences $g$ tending to infinity in the direction of $\eta$.

\begin{proposi}[{\cite[Proposition 2.3.2]{RabinovichRochSilbermann2004}}]
  Let $A\in {\mathcal L}({\mathbf X},{\mathcal P})$. Then $\sop(A) = \bigcup_{\eta\in S^{N-1}} \sop_\eta(A)$.
\end{proposi}

For $\eta\in S^{N-1}$ and $\omega\in\Omega$ define the \emph{limit set of $\omega$ at direction $\eta$} by
\[L^\eta(\omega):= \{ \nu\in\Omega :\; \text{$\exists\,(g_n)_{n\in{\mathbb N}}$ in ${\mathbb Z}^N$ tending to infinity in direction $\eta$ s.t. $\alpha(g_n)(\omega) \to \nu$}\}.\]

\begin{lemma}
  Let $\omega\in\Omega$. Then
  \[L(\omega) = \bigcup_{\eta\in S^{N-1}} L^\eta(\omega).\]
\end{lemma}

\begin{proof}
  We only have to show ``$\subseteq$, the other inclusion is trivial.
  Let $\nu\in L(\omega)$. Then there exists $(g_n)_{n\in{\mathbb N}}$ in ${\mathbb Z}^N$ such that $g_n\to\infty$ and $\alpha(g_n)(\omega)\to \nu$.
  Without loss of generality we may assume $g_n\neq 0$ for all $n\in{\mathbb N}$. Due to compactness of $S^{N-1}$, the sequence $(\tfrac{g_n}{|g_n|})$ has a convergent subsequence. Let $\eta$ be its limit.
  This implies $\nu\in L^\eta(\omega)$.
\end{proof}

\begin{proposi}
\label{prop:limit_sets} Let $\omega\in\Omega$ and $\eta\in S^{N-1}$. Then $L^\eta(\omega)$ is
non-empty, compact and invariant under $\alpha$.
\end{proposi}

\begin{proof}
Let $(g_n)$ in ${\mathbb Z}^N$ tend to infinity in direction $\eta$.
By compactness of $\Omega$, the sequence $(\alpha(g_n)(\omega))_{n\in{\mathbb N}}$ has a convergent subsequence.
Thus, the set $L^\eta(\omega)$ is non-empty.

\medskip

Let
$\nu=\lim_{n\to\infty}\alpha(g_n)(\omega)\in L^\eta(\omega)$, where $(g_n)_{n\in{\mathbb N}}$ tends to infinity in direction $\eta$.
Due to continuity, for $h\in{\mathbb Z}^N$ we have
$$\lim_{n\to\infty}\alpha(h+g_n)(\omega) = \alpha(h)\left(\lim_{n\to\infty} \alpha(g_n)(\omega)\right) = \alpha(h)(\nu).$$
Clearly, also $(h+g_n)_{n}$ tends to infinity in direction $\eta$.
Hence, $\alpha(h)(\nu)\in L^\eta(\omega)$ follows.

\medskip

We now show closedness of $L^\eta(\omega)$, implying that it is compact.
Let
$(\nu_k)_{k\in{\mathbb N}}$ in $L^\eta(\omega)$, $\nu_k\to \nu$. For
$k\in{\mathbb N}$ there exists $(g^k_n)_{n\in{\mathbb N}}$ tending to infinity in direction $\eta$ such that
$\nu_k=\lim_{n\to\infty} \alpha(g^k_n)(\omega)$.
Thus, for a suitable subsequence $(n_k)_{k\in{\mathbb N}}$ we obtain that $(g^k_{n_k})_{k\in{\mathbb N}}$ tends to infinity in direction $\eta$ and
$\lim_{k\to\infty}\alpha(g^k_{n_k})(\omega) = \nu$. Consequently,
$L^\eta(\omega)$ is closed.
\end{proof}

Note that in case $N=1$, we have $S^0 = \{-1,1\}$. Then $L^{+1}(\nu)$ is the so-called $\omega$-limit set and $L^{-1}(\nu)$ the $\alpha$-limit-set of $\nu$.

\begin{exam}[Random Schr\"odinger operators in $L_2({\mathbb R}^N)$] \label{ex:RandSchr}
  Let $Q:=(0,1)^N$ be the unit hypercube.
  Let $0\leq v\in L_\infty({\mathbb R}^N)$, $v|_{{\mathbb R}^N\setminus Q} = 0$.
  Let $l>0$ and equip $[0,l]^{{\mathbb Z}^N}$ with the product topology.
  Note that this topology is induced by the metric
  \[d(\omega,\nu):= \sum_{k\in{\mathbb Z}^N} 2^{-|k|} \min\{|\omega_k-\nu_k|,1\}.\]
  Let $\Omega\subseteq [0,l]^{{\mathbb Z}^N}$ be closed and translation invariant.
  Then $\Omega$ is compact and ${\mathbb Z}^N$ naturally acts on $\Omega$. Let $\alpha$ be this action.
  For $\omega\in\Omega$, define
  \[V_\omega:=\sum_{k\in {\mathbb Z}^N} \omega_k v(\cdot-k) \in L_\infty({\mathbb R}^N),\]
  and
  $H(\omega)$ in $L_2({\mathbb R}^N)$ by
  \begin{align*}
    D(H(\omega)) & := W_2^2({\mathbb R}^N),\\
    H(\omega) u & := -\Delta u + V_\omega u.
  \end{align*}
  Then $H(\omega) \geq 0$ holds for all $\omega\in\Omega$.

  Define $A(\omega):= (H(\omega)+1)^{-1}$. By \cite[Proposition 1.1.8]{RabinovichRochSilbermann2004}, a Combes-Thomas estimate \cite[Theorem 2.4.1]{Stollmann2001}
  and identifying $L_2({\mathbb R}^N) = \ell^2({\mathbb Z}^N,L_2(Q))=:{\mathbf X}$ we obtain $A(\omega)\in {\mathcal L}({\mathbf X},{\mathcal P})$ for all $\omega\in\Omega$ where $P_n:={\mathbf 1}_{\{-n,\ldots,n\}^N}$ for $n\in{\mathbb N}$.
  Furthermore, $\norm{A(\omega)}\leq 1$ holds for all $\omega\in\Omega$.
  By construction, $(i)$ is satisfied.
  Let us turn to condition $(ii)$, so let $(\omega^n)$ be a sequence in $\Omega$, $\omega^n\to\omega$.
  By the second resolvent identity it suffices to check that $\Plim_{n\to\infty}V_{\omega^n}=V_\omega$ as multiplication operators. Since $(V_{\omega^n})$ is uniformly bounded, we have to verify that
$$\lim\limits_{n\to\infty}\|P_m(V_{\omega}-V_{\omega^n})\| + \|(V_{\omega}-V_{\omega^n})P_m\|= 0 \quad (m\in{\mathbb N}) .$$
  Let $m\in{\mathbb N}$ and $\varepsilon>0$. Since $\omega^n \to\omega$, there exists $n_0\in {\mathbb N}$ such that $|\omega^n_k-\omega_k|\leq \varepsilon$ for all $n\geq n_0$ and $k\in\{-m,\ldots,m\}^N$.
  Hence, $\|P_m(V_{\omega}-V_{\omega^n})\| + \|(V_{\omega}-V_{\omega^n})P_m\|$ tends to zero if $n$ goes to infinity.

  Thus, if $(\Omega,{\mathbb Z}^N,\alpha)$ is minimal, then, the spectrum of $(A(\omega))_{\omega\in\Omega}$ is constant.
  Note that, for $\lambda\neq 0$, we have $\lambda\in \spec(A(\omega))$ if and only if $\frac{1-\lambda}{\lambda}\in \spec(H(\omega))$ (recall that $0\in \spec(A(\omega))$ since $H(\omega)+1$ is not bounded).
  Hence, we can also conclude constancy of the spectrum for the unbounded operator family $(H(\omega))_{\omega\in\Omega}$ with the help of our techniques.
\end{exam}

redpajama_set_name: RedPajamaArXiv


text:
Has anyone ever regretted getting a PhD? Do you feel like you missed out on real life or work experiences while pursuing a PhD? Are you haunted by the opportunity cost of studying for a PhD?
Updated on : January 21, 2022 by Ricardo Torres
I had a miserable time towards the end of my PhD research career. I had two advisers, one at the university and the other at a pharmaceutical company (3 hours away in another city) where I did my postgraduate research project.
One day my advisor, who rarely watched me, asked to see me in his office. It was at the end of the day, when most of the other scientists and employees of the company staff had left. I was about the sixth year of my PhD career, hated my life and hated scientific research, and constantly fantasized about quitting smoking and applying for a master's degree instead.
The first question my advice
The first question my advisor asked me was, "Jane, why are you doing this PhD?"
I was surprised by the question, not so much because of the frankness as because of my lack of response. No, not the rehearsed and canned answer about contributing to my field or seeking the truth or participating in this important quest for knowledge. I'm talking about the truthful answer as to why I was there.
It was obvious to me and to everyone who saw or spoke to me: I didn't want to be there.
The first thought that came to mind was, "I'm doing this for my dad."
My father was the first high school student from his hometown to be accepted into the prestigious National Taiwan University. Until then, students either became farmers like their parents or went to receive vocational training. Everyone said I was smart like my father, although to be honest, I didn't have the caliber of student that he was. My father was the scholar, the best student, the teacher of the family.
One of my father's regrets had always been to dwell only on master's degrees (he earned 2: Master of Economics from National Taiwan University and Master of Engineering from Stanford University). He would have gone on to a doctorate, but he married and had me and my brother. He needed to get "a real job" and to work and support the family.
Somehow I decided that I would get a PhD and it would be the first PhD in my family, and it would make him proud. Maybe doing it would help a little to make him feel like he had done it. Growing up, he had been her favorite son and people said we were "so alike."
My advisor told me that I needed to do this PhD myself. Not for my dad or anyone else. This made complete intellectual sense, of course.
Except for this point, I was too far from changing my destiny and never becoming a PhD student in the first place, and I was not far enough away in my research to go out there and not feel like I had wasted 10 years of my life blindly. Walking down a path that I had never questioned Scientists are supposed to question everything in search of the truth. I did not apply that search for truth to the kind of life I wanted to create.
I was never taught to think of "my life" as "mine"; the purpose of my life was primarily to make my parents proud. As I distanced myself from my parents during my teens and youth, what "made my parents proud" mattered less and less. Surviving clinical depression (including a misdiagnosis that led me to take an antiepileptic drug that nearly killed me) mattered more. Spending each day alive mattered more.
What changed me was meeting my husband, because now I had a personal life goal: I needed to graduate so I could finally move from New York to California and be with him. He had already told his roommate to move out because he was anticipating that I would move out. Basically, I reoriented my thinking from wanting to quit to motivating myself to quit. The man needed me to finish, get a job, and help pay the rent, for crying out loud! This is California!
I'm not going to repeat what I wrote here about how it feels to complete my final Ph.D. defense and whether pursuing a Ph.D. is really worth pursuing, especially when I was one of those who dropped out of academia after graduate school or the pros and cons. the cons of getting a degree. PhD if I have no interest in staying at the academy. You can get a good idea of whether I felt my PhD was worth doing, despite the misunderstood sense of filial duty that made me choose the path in the first place.
I remember getting my dissertation box: copies of the bound volume of 7 years of work that mostly came to nothing, both through no fault of my own and my own poor choice, and the value of 1 year of work that actually did a bit of knowledge coherent enough to be worth writing and binding in black hardcover.
Yes, the title is a mouthful. 2 copies of the dissertation are in this part of our shelf where all my writings are, including the books I have published since then.
I sent a copy of my doctoral thesis to my dad.
We were still apart but… why not.
One night I called him. "Dad, I got my PhD. Aren't you proud of me?
He said, "An MBA is more practical."
Like I said, we were separated.
After hanging up, I waited a long time until I wanted to talk to him again.
Perhaps it hurt him to see that his name was not mentioned on the cover, where I thanked the people who were there for me during the PhD process. My two advisers who believed in me. My husband, who told me that quitting was not an option. A friend in the lab who always came to let me know where there was free food from a conference or meeting.
Even if I had originally taken the Ph.D. path to please him, it was I who earned the badge after my name.
Perhaps the honor would have been transferable if the love between us hadn't been broken and broken over the years.
I started doing the doctorate for my dad, I ended up doing the doctorate on my own. This alone made the effort of doing my Ph.D. worth it forever.
Well, in my experience, there are two types of people who want to get a PhD:
- Those who really wanted to investigate and knew what it meant. But you really have to take research as a vocation. It's a bit like being a priest. If you don't worship science for science's sake, you probably won't get any satisfaction from it. Worse still, this is an extremely competitive field, where most people will only prosper (who are the best of the best). If you can't advance to the state of the art, you will likely end up a lab monkey, doing experiments for others. And the pay will suck. Please unde
- Those who really wanted to investigate and knew what it meant. But you really have to take research as a vocation. It's a bit like being a priest. If you don't worship science for science's sake, you probably won't get any satisfaction from it. Worse still, this is an extremely competitive field, where most people will only prosper (who are the best of the best). If you can't advance to the state of the art, you will likely end up a lab monkey, doing experiments for others. And the pay will suck. Please understand that all people who earn a PhD are already very good and only people who are very dedicated to their science have the opportunity to make a difference. And yes, politics matters in the lab, but politics alone won't save you.
- However, many people choose a doctorate because they want a diploma that is considered the highest they can obtain. This is very dangerous for them and usually a waste of time. Labs need people to do experiments and run what the stars want. So they won't reject you unless you're bad. But this does not mean that you have the level to be a good investigator. Doing experiments at night for others will not make you a good researcher. Being very knowledgeable about your field (above everything you learned in college) and being very curious is what will make you a good researcher. You will need not to have any dogma about anything, you will have to question everything, and you will need to enjoy theory because it is the language in which you work to gather the truth about the world.
If you never opened a book on your subject yourself (and not because you were assigned to do so). If you don't wake up at night thinking about cool experiments you'd like to do, then don't go the Ph.D. route. It's not for you. If you are like this, work for the private sector. However, you will benefit from an internship in a good laboratory.
I don't have a PhD myself, but I worked as a research engineer in a laboratory setting. It was full of really brilliant people and since I was an engineer and not a researcher, I wasn't supposed to do research but rather develop systems to help with research. However, I had the opportunity to make good contributions and felt useful there. I even co-wrote posts with my boss and we made a pretty good team. In that sense, I had a better time there than some PhD students (but not all). After that, I entered the corporate world, working for various companies. What I learned in the laboratory in terms of methodology and scientific knowledge was very valuable. But in the lab there were a lot of students (especially Asians brought from other countries or raised by tiger mothers) who were there to get their diploma and they weren't very passionate about the field. Everyone in the lab knew they had no future in research, but they were dedicated, cheap (thanks to the Ph.D. program), and made all the work not good enough to be done by top brass. If I could meet these job seekers now, I don't think I would hire them. They didn't really benefit from the research environment there. They are workers, but life is not about who works hard, but who gets results. If I could meet these job seekers now, I don't think I would hire them. They didn't really benefit from the research environment there. They are workers, but life is not about who works hard, but who gets results. If I could meet these job seekers now, I don't think I would hire them. They didn't really benefit from the research environment there. They are workers, but life is not about who works hard, but who gets results.
NB: Don't misunderstand my comments about previous Asians, but they were culturally much more inclined to do something they don't like but are good at than other people. It's part of your education (look at how education works in China (or it used to work because now if you're a Fuerdai it's not like that anymore)). On the other hand, both whites and blacks generally won't be able to put in the level of effort required to get into a Ph.D. program if they don't like the subject. This avoids the problem. At some point in your maturity, knowing what you love and what you want to do is an important milestone. Which is even more important than any degree you can get. If you don't even have a Ph.D. it won't save you. It will make you more miserable.
I am currently a postdoc in an experimental science field, and just a few days ago I decided that not only am I going to leave academia, but it is more than likely that I will leave the field in which I obtained my doctorate. This decision essentially makes most of my technical knowledge and all of my formal academic achievements (minus the degree itself) moot, and I agree with that. My reasons for wanting to get out are pretty simple: staying on this path would make my life surprisingly predictable (and pretty monotonous), I'd have to hyperspecialize at the expense of many other ideas and opinions.
I am currently a postdoc in an experimental science field, and a few days ago I decided that I am not only going to leave academia, but that I will more than likely leave the field in which I obtained my PhD. This decision essentially makes most of my technical knowledge and all of my formal academic achievements (minus the degree itself) moot, and I agree with that. My reasons for wanting to get out are pretty simple: staying on this path would make my life surprisingly predictable (and pretty monotonous), it would require me to hyper-specialize at the expense of many other ideas and possibilities, and all of this would be taking place in context. intense competition for both funding and jobs. In short, I think my life would be the worst combination of stressful and bland,
Even though I am leaving behind my academic achievements (of which there are many, don't think I'm leaving because I can't do research) and much of my accumulated body of knowledge, I don't regret my PhD at all. Sure, much of my life was devoted to my studies (there was a time in graduate school where I had spent a quarter of my life working on my PhD), and choosing to focus on that deprived me of many opportunities, but I got A LOT out of my PhD. Among the benefits:
I learned to be adequately critical of everything.
I developed a strong sense of intellectual self: I know what I believe, I know why I believe it, and I have enough faith in those beliefs to be able to convince other people of them.
I learned to communicate effectively in a wide spectrum of media: written words, visual presentations, public speaking, informal conversations.
I convinced myself that I am capable of designing, executing and finishing complicated long-term projects.
I learned to teach myself almost anything, both skills and knowledge.
I made great strides (but I'm not done; I don't know if this is NEVER done) to find out who I am and why I do the things I do.
So even if you finish a PhD and then leave academia and your field immediately, there are many incredibly important and useful skills (both for your future professional and personal life) that you will have learned. Sure, I could have gone out and landed a "real" job straight out of college, but there's no way I would have earned my intellectual independence in the process. I could have had many "other" experiences, but I probably wouldn't have had the solid personal framework to understand and integrate them into who I am if I hadn't gotten my PhD.
I'm about to completely reshape my life and live it according to what I think matters the most (which differs quite a bit from conventional social values), and I don't think I could take on such a big challenge if I had. I didn't grow up over the course of my PhD.
I wouldn't recommend one for the vast majority of people, but I absolutely believe that my PhD was the right thing for me.
I definitely regret getting a PhD. I got a Ph.D. in organic chemistry over a year and a half ago, and I've been unemployed ever since; I haven't been able to get a job since I graduated. Compared to my peers, I feel that I have not achieved anything in life; I don't have a job, I have little savings, I don't have a car, family, wife or girlfriends, much less children. I go back to living with my parents. I applied for over 2000 positions in a wide variety of areas - pharmaceuticals, materials, analytics, consulting, post-doc positions, and even bachelor's positions in despair - only to receive
I definitely regret getting a PhD. I got a Ph.D. in organic chemistry over a year and a half ago, and I've been unemployed ever since; I haven't been able to get a job since I graduated. Compared to my peers, I feel that I have not achieved anything in life; I don't have a job, I have little savings, I don't have a car, family, wife or girlfriends, much less children. I go back to living with my parents. I applied for over 2000 positions in a wide variety of areas - pharmaceuticals, materials, analytics, consulting, postdoc positions, and even bachelor positions in desperation - only to receive the same generic rejection email in my inbox. entry. weather. I sacrificed the peak years of my life (my 20s) on the altar of science - to get, well ... nothing.
Unfortunately, society sends mixed messages to PhD students. On the one hand, there are people who say "Wow, doing a PhD is great, you can change the world!" But once you graduate, you see the true value of the degree, which is ... less than toilet paper, due to the insane saturation of the market in both academia and industry. Another problem is that it is very difficult to find employment statistics for graduates of doctoral programs; This data is crucial to be able to assess the relative strength of a program, because after all, you get a degree to get a job and make money, right? But most colleges don't care what happens to their graduates after they get a Ph.D., which is very unfortunate.
I missed out on a lot of real life experiences due to the high opportunity cost of a PhD in organic chemistry - I didn't have time to get a girlfriend or have any kind of relationship because I was too busy working in the lab all the time. I am almost 30 years old and have never had a full time job. The other problem is that a Ph.D. in organic chemistry leaves you with very few "transferable skills," making it extraordinarily difficult to get jobs outside of chemistry (and keep in mind that getting a job in the chemical industry with a Ph.D. chemistry is almost impossible).
In a word, yes, but it is a complicated yes.
To begin with, he had no great reason to go to graduate school; it was essentially a combination of overblown tendencies, indecision, and a bad job market. I graduated in 2010 from Boston University with my bachelor's and master's degrees in economics through a joint degree program, but the recession created a tough job market. However, I loved the statistics courses I took at the end of my program and was really interested in the health insurance markets (Obamacare was turning into something). I thought a PhD in biostatistics would be a good way to grow my
To begin with, he had no great reason to go to graduate school; it was essentially a combination of overblown tendencies, indecision, and a bad job market. I graduated in 2010 from Boston University with my bachelor's and master's degrees in economics through a joint degree program, but the recession created a tough job market. However, I loved the statistics courses I took at the end of my program and was really interested in the health insurance markets (Obamacare was turning into something). I thought that a PhD in biostatistics would be a good way to develop my skills and wait for the economy to pass.
Not everything went as planned. My research area changed based on the availability of advisers (from public health to bioinformatics), and my partner's graduation at 3 years of my career meant that we moved across the country. I started flying from California to Massachusetts to continue my thesis, alternating between being on the other side of the country from my institution and from my husband. At the end of my fourth year in the PhD program, I applied for internships at a combination of biotech and technology companies, and was fortunate to receive an offer from Quora. After a summer internship with the data science team there, I never looked back. I finished my PhD, but knew that I would not go back to academia (not even biology) after graduating.
Grad school was a difficult time in my life. Sometimes the best I can say is that it broke me in helpful ways; More than anything, I learned to work much harder than ever. It's hard for me to put the label "regret" on something that taught me so much about myself ... but then I think of everything I've learned in just over a year of full-time work. I can't help but feel like I would have grown so much more if I hadn't spent 5 1/2 years studying a degree that I hardly use.
As it stands, graduate school was an extremely inefficient way to get to a place where I am actually really happy. I don't regret the person who made me, but it's hard to calculate all the costs and believe that this was really the best option.
There are tons of people who would have regretted getting a PhD for various reasons. Some feel the title doesn't live up to the hype of calling itself "Doctor." Some feel that if they had remained in the "industry" they probably would have received a higher salary and a higher position. Some would have had terrible experiences with doctoral advisers in inhumane and voiceless conditions. Some may become depressed due to poor results or because they find that they lack the ability to conduct original basic research. Others may find that they do not like academic life and more e
There are tons of people who would have regretted getting a PhD for various reasons. Some feel the title doesn't live up to the hype of calling itself "Doctor." Some feel that if they had remained in the "industry" they probably would have received a higher salary and a higher position. Some would have had terrible experiences with doctoral advisers in inhumane and voiceless conditions. Some may become depressed due to poor results or because they find that they lack the ability to conduct original basic research. Others may find that they dislike academic life and enjoy solving real-world problems more. All of these are fine. A PhD is not for everyone. In my humble opinion, A good PhD is one in which you develop the ability to think critically about a problem, generate multiple testable hypotheses and test them rigorously while eliminating subjective bias as much as possible. He does this through continuous practice for 6 to 7 years.
My former advisor said that the best doctors are those who had no idea what they were doing and graduated because those people were capable of being independent thinkers and entrepreneurs. Now, whether those skills are learned or innate is another question. This is what you should expect to gain from joining a good PhD program, the ability to think critically about whatever problem you want. If you don't want that skill or already have it, don't bother, do something more useful with your time. The rest, the articles, the experiments, the post counts are just numbers that people use to justify their existence. Also, I believe that a PhD is just one of many experiences in your life. While it may be a terminal degree, it is not a terminal experience, there are many more things to look forward to and do. A PhD is something that just happens while doing something you love. As I am discovering, it is the same for startups, postdocs, chairs, and anything else in life. It is a piece of paper that you get along the way.
DO NOT join a PhD for the glory of a degree. DO NOT join him expecting a big pay later. You get a great salary if you are a stellar researcher or an excellent communicator or a great thinker, which again are traits that you can develop through continuous learning and practice (during or without a PhD).
I have a PhD in Modern European History. He was a specialist in the British Empire and revolutions. I taught at the City University of New York and Lafayette College.
And I've spent the last twenty-five years working on Wall Street.
Today, I would never recommend anyone to study a doctorate in history. It is nothing but exploitation and poverty, with very little chance of getting a permanent job. It is also extremely difficult to transition out of academia, certainly much more so now than when I did in the mid-1990s, so depending on how old you are when you finish, if you start a PhD program
Today, I would never recommend anyone to study a doctorate in history. It is nothing but exploitation and poverty, with very little chance of getting a permanent job. It's also extremely difficult to get out of academia, certainly much more so now than when I did in the mid-1990s, so depending on how old you are when you finish, whether you start a humanities doctoral program or many of the In the Today, in the social sciences, with a few exceptions such as economics and psychology, you may be condemning yourself to a life of poverty and a lack of meaningful work.
Now that that's clear, I don't regret a minute of my time in graduate school. I have no regrets about getting a doctorate in history and, in fact, I am quite proud of my thesis. I studied with one of the best living British historians in the world, David Cannadine, and I remain friends with him all these years after. In fact, if it weren't for the coronavirus, we'd be meeting for dinner right now.
I also met my wife when I was in graduate school. That was thirty-two years ago, two great kids and two great dogs.
I didn't "miss out on real life experiences" because getting an education is a real life experience. Teaching while studying is a real life experience. Living close to poverty is a real life experience. No "opportunity cost" haunts me either. It certainly cost me a lot in terms of lost income and student loan debt, but I paid for it all and made a lot of money afterward.
How did i do that? Because a doctorate in history provides a phenomenal foundation for studying markets. I was the top fixed income salesperson in my last store, which was part of one of the largest banks in America. I raised $ 25 billion for them in eleven years, with $ 8 billion in 2008 alone and $ 4.5 billion in 2009. The historian. Not the people in the company who studied economics and finance. They all lost money in 2008.
My chief investment officer once asked me how I managed to perform so well in asset management with such an untraditional background. So I told him. Historians are trained to take an enormous amount of disparate data and synthesize it into a coherent narrative. You do the same when you study and explain the markets. Historians are experts at taking that narrative and helping students understand the story. The same goes for investors trying to understand markets, especially markets in crisis. Also, historians are either blessed or cursed (it depends on how you look at it), with very long-term prospects. I can go back to the Roman Empire if I find the right analogy for a client and help them understand in ways that make perfect sense to them.
All of this is to say that PhDs have enormous skills that are easily transferable to many other fields, but that the business world doesn't really understand beyond quantitative disciplines. When I was working in a hedge fund fund at a major American bank, there were five doctors on the team, including the boss, but mine was the only one who was not quantitative. I used to say that they were the ones who did all the real work understanding correlations and portfolio and risk building methodologies, and I was the PhD who explained what they were doing consistently to our clients.
So no, I don't regret a minute I spent in graduate school. But I have advised my children not to follow in my footsteps unless they obtain a Ph.D. in a field where there are clearly defined employment opportunities outside of academia. Look for the opportunities of the 21st century, not the 20th.
So AI or alternative energies, sure, get a PhD. History or classics, no. Which is really a shame. The world needs a better understanding of topics like history and classics. It would be a much better place if people studied these fields. But they don't, and you can't make a living chasing them anymore.
And don't go to Wall Street either. But that's another topic for another day.
I left a doctoral program in food sciences and am planning to start a doctorate in French literature.
The key is to know why you are doing the PhD. Sorry for the Ph.D. attempt in food science. I got into it not because I wanted to do research, but because I missed school and wanted to teach. As a person socially uncomfortable with teaching and counseling as career goals, I saw a PhD as a means to an end, a way to use my academic strengths rather than my weaknesses in interpersonal skills to land a job that, however, would allow me to do. everything I wanted to do. Research was simply something I had
The key is to know why you are doing the PhD. Sorry for the Ph.D. attempt in food science. I got into it not because I wanted to do research, but because I missed school and wanted to teach. As a person socially uncomfortable with teaching and counseling as career goals, I viewed a PhD as a means to an end, a way to use my academic strengths rather than my weaknesses in interpersonal skills to land a job that, however, would allow me to do. everything I wanted to do. The investigation was simply something that I had to put up with. In fact, I had applied to Ph.D. programs in food science, comparative literature, and education, depending on the school I applied to. I didn't care what I was studying
I was wrong. I avoided the lab and unconsciously saw it as a waste of time. I became irritable, annoyed, and impatient. I turned to other activities to stand my ground and give myself a sense of purpose: chorus, teaching, student leadership, and my simultaneous mastery. The laboratory. This whole experience cost me my emotional well-being and my relationship with various teachers, and for that I regret it. I also regret spending three years doing things I didn't enjoy toward a goal that didn't really lead to the career I wanted ... when there are so many things I could have been doing that would have been worthwhile.
However, I will not regret the program I plan to enter this fall. And I say this knowing full well that entering PhD students rarely know if they will finish, and I say this knowing full well that graduating PhD students rarely find the teaching position they want. The thought of spending many more years in school doesn't scare me as much as when I was in food science. Why? Because I'm in it for the right reason this time. Because I'm actually curious, because my future research and that of others really excites me and keeps me focused, even if the work itself can be exhausting. I want to be a teacher. The job of a professor is research, teaching and service. The work of a doctoral student is also research and teaching, with the opportunity to participate in the service. Why would these five years be a waste of time? when it allows me to do everything I want to do in a career and get paid for it? If this job lasts only these five years, so be it. It's better than nothing. I'm not missing out on real life experiences because these are my real life experiences. They are exactly what I will do in my ideal career, and they should be appreciated even more because I don't know where I will end up next. I'm not missing out on real life experiences because these are my real life experiences. They are exactly what I will do in my ideal career, and they should be appreciated even more because I don't know where I will end up next. I'm not missing out on real life experiences because these are my real life experiences. They are exactly what I will do in my ideal career, and they should be appreciated even more because I don't know where I will end up next.
I do not regret my PhD (obtained in 1978… at different times). I can understand why many recent doctors may regret it.
In many academic fields, the supply of PhDs far exceeds the demand. People hoping to get permanent academic jobs are often disappointed (only about 25% of university professors in the US are now in permanent positions; universities find it much cheaper to hire people per course, adjunct, lecturer, instructor or in soft courses). money research positions for which the person must generate their income through grants).
Depending on the field of study, there can be many job opportunities.
Depending on the field of study, there can be many employment opportunities outside of academia. But for some fields, that is not the case (comparative literature, for example).
The worst case scenario is that a person incurs tens of thousands of dollars (or other currencies) in student loan debt and spends 5 or more years of life to obtain a degree, and then discovers that the job prospects are poor.
If you can get a tuition waiver and stipend, and have a genuine love for the field of study and choose a department that is well suited, you can get a lot out of the PhD process, even if it doesn't lead directly to the career and earnings you hope for. .
In my field, when the department advertises a permanent position, there are dozens and sometimes even hundreds of applicants. Many of these people, sometimes even at the new PhD level, have extensive publications, strong teaching experience, and grants. To get a job, you need exceptional qualifications, but sometimes they are not enough. Often times, people also need luck, that their research interest matches exactly what a department is looking for, and that that department is hiring when that person is on the job market.
I can understand very well why some people with PhDs feel that there could have been better ways to spend their money and time. I believe that when people apply for PhD programs, they should be told what kind of jobs recent graduates have found.
It tears my heart to admit it, but I think it's better to deal with the truth about yourself and your life, no matter how much it pissed you off.
Before starting my PhD, I had just identified the existing theory that I really wanted to understand as much as possible (and was dreaming of subsuming it by introducing a theory of my own). However, I ended up working on a different theory for my PhD. One that I was not that interested in, that I was not well prepared to deal with, and that was also much less fashionable all over the world (within the math community).
I have not got
I have not yet obtained my PhD, but I know that I will always have great regrets for having dedicated 5 years of my life (most importantly: my youth) to it.
Why ? The reasons are many ...
Because I didn't like living where I did my PhD
Because I wasted my time
Because I suffered for nothing but a title
Because I wasted my energy
Because I lost my trust
Because I lost many contacts in the world mathematical community that I left disappointed (they expected me to produce a very good dissertation)
Because I did not learn so many things, much less things that I wanted to know deeply.
Because it kept me from delving into the theory I was supposed to probe, when was the best time to do so.
Because I lost my scientific pride
Because it fills me with insecurities and complexes: I feel uncomfortable and incompetent among my friends who had a better destiny as PhD students (many of whom will probably end up being university professors).
Because I was not paid for almost two years of my work on my thesis, the last
Because there is no postdoctoral fellowship for me: the end of my doctorate means the end of my academic life.
Because I can no longer dream of becoming a great scientist
Yes, after weighing all the pros and cons, I definitely regret getting my PhD. When I started, I had a "romantic" idea to pursue science. Everyone in science is working together to make this world 'a better place', right? or not?!? Oh boy, how could I be so wrong! All the funding scramble, treacherous colleagues, nepotism, unhealthy competition, time pressure, etc., is not really a pleasant environment to spend much of your life in. After four years earning my PhD and numerous postdocs later, I can say one thing; don't do it unless you're 200% motivated and have an almost infinite passion for d
Yes, after weighing all the pros and cons, I definitely regret getting my PhD. When I started, I had a "romantic" idea to pursue science. Everyone in science is working together to make this world 'a better place', right? or not?!? Oh boy, how could I be so wrong! All the funding scramble, treacherous colleagues, nepotism, unhealthy competition, time pressure, etc., is not really a pleasant environment to spend much of your life in. After four years earning my PhD and numerous postdocs later, I can say one thing; don't do it unless you're 200% motivated and have an almost infinite passion for doing it. When I was 40 I left the academic world, realizing that this was not the life I wanted and did not imagine when I started. The academic career possibilities are very scarce, always with short temporary contracts, mediocre salary, competition is fierce, research funding is very difficult. PhD research positions are flooded with idealistic young students willing to work hard in the hope of succeeding in science (as I once did). Not realizing that the influx of PhD students is roughly ten times greater than the long-term positions available in science. PhD research positions are flooded with idealistic young students willing to work hard in the hope of succeeding in science (as I once did). Not realizing that the influx of PhD students is roughly ten times greater than the long-term positions available in science. PhD research positions are flooded with idealistic young students willing to work hard in the hope of succeeding in science (as I once did). Not realizing that the influx of PhD students is roughly ten times greater than the long-term positions available in science.
Everything is wrong? No, you are likely to become an independent and self-reflective person with a lot of intellectual baggage. What struck me the most in the end; a doctorate in the world outside of academia is not worth much. It would even go so far that it goes against you when you look for work in the 'normal' world ...
I still love science!
During my academic years I also met a lot of lovely and caring people ...
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What professions are overpaid or underpaid?
Which type of job can I get in Chennai as part time job at night?
Why am I nervous about a job and an interview?
¿A qué temerán los partidarios de Trump si pierde la reelección en 2020?
Is there a way to do a PhD without quitting my job?
What are the chances of getting an IT job in Australia?
Why does a civil engineer have less salary? I am 22 years old with 1 year of experience and a salary of 17k. I hardly get a 1 or 2k increase per year. At this rate, it would take more than 5 years to reach 25k, which in itself is not enough.
What jobs are suitable for someone who cannot speak English?
Why is it difficult to get a good job in the UK?

redpajama_set_name: RedPajamaCommonCrawl


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text:
Two Books: Charles Martin, Future Perfect; Adam Tavel, Catafalque
by Brooke Clark | Print Edition - Winter 2018
| Tags: Book Reviews adam tavel author book review brooke clark catafalque charles martin future perfect new criterion prize v26 winter 2018
Brooke Clark
Charles Martin, Future Perfect
Baltimore, Maryland: Johns Hopkins University Press, 2018
ISBN 978-1-4214-2535-1, 88 pp., USA $19.95, paperback
Adam Tavel, Catafalque
Evansville, Indiana: The University of Evansville Press, 2018
ISBN 978-0-930982-77-5, 108 pp., USA $15.00, hardcover
Charles Martin's new collection, Future Perfect, is a book deeply involved with books. From a translation of a passage of Euripides, to a poem based on a Biblical story, to an imagined letter from Anna Akhmatova to Robert Frost, to several poems drawn from the life and work of Weldon Kees, virtually all the poems in this book are engaged in dialogue with other texts. This gives the collection a reverberant quality, as our own memories of, or ideas about, the texts Martin is responding to echo through our reading of his poems, making Future Perfect in some ways more than the sum of its parts. As well as being a tour through texts that have meaning for its author, Future Perfect could almost be considered an encyclopedia of forms. Its seventy pages include sonnets, villanelles, sestinas, and poems written in ottava rima stanzas, quatrains and couplets, among others. The forms are all expertly managed, and the range provides a pleasing sense of variety as one goes through the collection.
In addition to stand-alone short lyrics, the book contains several longer sequences. "From Certain Footprints Found at Laetoli," inspired by a news article, is a . . .
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print edition :: book review
Able Muse: Print Edition, Winter 2018
A Review of Ange Mlinko, Distant Mandate
Two Books: A review of Adelaide Crapsey - On the Life and Work of an American Master and Jiggery-Pokery Semicentennial
Two Books: Aaron Poochigian, Mr. Either/Or; Rebecca Foust, Paradise Drive
Two Books: X. J. Kennedy, That Swing: Poems, 2008–2016; Alexandra Oliver, Let the Empire Down


text:
Anisopodus cognatus is een keversoort uit de familie van de boktorren (Cerambycidae). De wetenschappelijke naam van de soort werd voor het eerst geldig gepubliceerd in 1863 door Bates.

cognatus

redpajama_set_name: RedPajamaWikipedia


text:
Basic information Name: Bumble Bee
Meteoritical Bulletin: MB 104 (2015) H6
Comments: Approved 14 Dec 2015
Bumble Bee 34° 12' 00"N, 112° 09' 45"W
Find: 2004
Classification: Ordinary chondrite (H6)
History: Around the spring of 2004, Mr. Ed Bishop found an unusual rounded brown stone while he was metal detecting for gold, ~1/2 mile west of the ghost town of Bumble Bee, Arizona. He brought the stone to Carleton Moore at ASU, who subsequently purchased it.
Physical characteristics: Well-rounded stone covered in patches of remnant fusion crust. The stone has a shape of a squat teardrop and lacks regmaglypts. Interior is a dark gray-green with a uniform distribution of small metal and troilite grains. Chondrules not visible.
Petrography: (L. Garvie, ASU) Thin section shows largely recrystallized texture with only two recognizable chondrules (both BO, 400 and 450 μm diameter). Olivine grains show straight extinction characteristic of S1. Plagioclase abundant and coarse grained, to 200 μm. Troilite (<200 μm) is single crystal and lacks shock lamellae. Metal dominated by kamacite (<250 μm) and zoned grains with tetrataenite rims and kamacite cores (<150 μm). Only one Cu grain (40 μm) found between troilite-metal grains. Chromite to 200 μm shows rounded outlines and lacks cracks. Scattered phosphates (Ca-Cl and Ca-Mg-Na) to 200 μm.
Geochemistry: (L. Garvie, ASU) Olivine, Fa19.5±0.2, Fe/Mn=38.9±2.1, n=11. Low Ca pyroxene Fs17.4±0.2Wo1.2±0.2, Fe/Mn=23.1±1.5, n=10.
Classification: H6, S1, W1
Specimens: 41 g and one thin section at ASU.
Latitude: 34° 12' 00"N
Longitude: 112° 09' 45"W
Mass (g): 348
Class: H6
Ferrosilite (mol%): 17.4±0.2
Wollastonite (mol%): 1.2±0.2
Classifier: L. Garvie
Type spec mass (g): 41
Type spec location: ASU
Main mass: Carleton Moore
Finder: Ed Bishop
Comments: Submitted by L. Garvie
and collections ASU: Center for Meteorite Studies, Arizona State University, Tempe, Arizona 85287-1404, United States; Website (institutional address; updated 14 Jan 2012)
References: Published in Meteoritical Bulletin, no. 104, MAPS 52, 2284, Octover 2017, http://onlinelibrary.wiley.com/doi/10.1111/maps.12930/full
Recommended:: (34° 12' 0"N, 112° 9' 45"W)


To develop AmCham into a leading organization promoting CSR principles and best practices within the Moldovan business community.
The Corporate Social Responsibility Committee is designed to encourage good Corporate Social Responsibility amongst AmCham Members and the wider business community in general, with the vision of developing AmCham into the leading organization in Moldova promoting CSR principles and best practice. Increasing awareness of CSR and a company's obligations as a part of this is one of the key activities of the committee. The Committee is also responsible for encouraging companies to make a positive impact through their activities on the environment, consumers, employees, communities and stakeholders.
Want to take part in the CSR Committee? Contact committee Coordinator Elena Buzu at elenabuzu@amcham.md for more information.
Committee members participate in Committee events and meetings, take active role in Committee's activity, undertake common projects.
CSR Committee will meet on the premises of member companies on a rotating basis or at AmCham, for Committee meetings, roundtable discussions on various issues related to CSR. The Committee may also envisage inviting outside speakers, if appropriate.
The key directions of committee work are information exchange - increase the public awareness and business to business communication, best practice share - improve the performance of CSR policies and operations in/between the key stakeholders (government bodies, businesses) and implementation of common projects.
Committee coordinator will regularly inform members on Committee activity and will also contribute to AmCham Weekly Bulletins, Annual Reports, and Web site.
The Committee is open to individuals who have occupation relevant to the Committee's stated mission and whose companies are members of AmCham. We kindly invite every interested member to join our efforts.
In that spirit, no Committee member, including Co-Chairs, shall express personal views on behalf of the Committee and therefore AmCham, unless they represent an official and approved position by the Board of Directors.