Unbounded operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since
 "unbounded" should sometimes be understood as "not necessarily bounded";
 "operator" should be understood as "linear operator" (as in the case of "bounded operator");
 the domain of the operator is a linear subspace, not necessarily the whole space;
 this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense;
 in the special case of a bounded operator, still, the domain is usually assumed to be the whole space. In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain. The term "operator" often means "bounded linear operator", but in the context of this article it means "unbounded operator", with the reservations made above. The given space is assumed to be a Hilbert space. Some generalizations to Banach spaces and more general topological vector spaces are possible. Short history
The theory of unbounded operators developed in the late 1920s and early 1930s as part of developing a rigorous mathematical framework for quantum mechanics. The theory's development is due to John von Neumann and Marshall Stone. Von Neumann introduced using graphs to analyze unbounded operators in 1932. Definitions and basic properties 
Let  be Banach spaces. An unbounded operator (or simply operator)  is a linear map  from a linear subspace —the domain of —to the space . Contrary to the usual convention,  may not be defined on the whole space . An operator  is said to be closed if its graph  is a closed set. (Here, the graph  is a linear subspace of the direct sum , defined as the set of all pairs , where  runs over the domain of  .) Explicitly, this means that for every sequence  of points from the domain of  such that  and , it holds that  belongs to the domain of  and . The closedness can also be formulated in terms of the graph norm: an operator  is closed if and only if its domain  is a complete space with respect to the norm:

 

An operator  is said to be densely defined if its domain is dense in . This also includes operators defined on the entire space , since the whole space is dense in itself. The denseness of the domain is necessary and sufficient for the existence of the adjoint (if  and  are Hilbert spaces) and the transpose; see the sections below. If  is closed, densely defined and continuous on its domain, then its domain is all of . A densely defined operator  on a Hilbert space  is called bounded from below if  is a positive operator for some real number . That is,  for all  in the domain of  (or alternatively  since  is arbitrary). If both  and  are bounded from below then  is bounded. Example 
Let  denote the space of continuous functions on the unit interval, and let  denote the space of continuously differentiable functions. We equip  with the supremum norm, , making it a Banach space. Define the classical differentiation operator  by the usual formula:

 

Every differentiable function is continuous, so . We claim that  is a well-defined unbounded operator, with domain . For this, we need to show that  is linear and then, for example, exhibit some  such that  and . This is a linear operator, since a linear combination  of two continuously differentiable functions  is also continuously differentiable, and

The operator is not bounded. For example,

satisfy

but

as . The operator is densely defined, and closed. The same operator can be treated as an operator  for many choices of Banach space  and not be bounded between any of them. At the same time, it can be bounded as an operator  for other pairs of Banach spaces , and also as operator  for some topological vector spaces . As an example let  be an open interval and consider

where:

Adjoint 
The adjoint of an unbounded operator can be defined in two equivalent ways. Let  be an unbounded operator between Hilbert spaces. First, it can be defined in a way analogous to how one defines the adjoint of a bounded operator. Namely, the adjoint  of  is defined as an operator with the property:

More precisely,  is defined in the following way. If  is such that  is a continuous linear functional on the domain of , then  is declared to be an element of  and after extending the linear functional to the whole space via the Hahn–Banach theorem, it is possible to find some  in  such that

since Riesz representation theorem allows the continuous dual of the Hilbert space  to be identified with the set of linear functionals given by the inner product. This vector  is uniquely determined by  if and only if the linear functional  is densely defined; or equivalently, if  is densely defined. Finally, letting  completes the construction of  which is necessarily a linear map. The adjoint  exists if and only if  is densely defined. By definition, the domain of  consists of elements  in  such that  is continuous on the domain of . Consequently, the domain of  could be anything; it could be trivial (that is, contains only zero). It may happen that the domain of  is a closed hyperplane and  vanishes everywhere on the domain. Thus, boundedness of  on its domain does not imply boundedness of . On the other hand, if  is defined on the whole space then  is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space. If the domain of  is dense, then it has its adjoint  A closed densely defined operator  is bounded if and only if  is bounded. The other equivalent definition of the adjoint can be obtained by noticing a general fact. Define a linear operator  as follows:

Since  is an isometric surjection, it is unitary. Hence:  is the graph of some operator  if and only if  is densely defined. A simple calculation shows that this "some"  satisfies:

for every  in the domain of . Thus  is the adjoint of . It follows immediately from the above definition that the adjoint  is closed. In particular, a self-adjoint operator (meaning ) is closed. An operator  is closed and densely defined if and only if 

Some well-known properties for bounded operators generalize to closed densely defined operators. The kernel of a closed operator is closed. Moreover, the kernel of a closed densely defined operator  coincides with the orthogonal complement of the range of the adjoint. That is,

von Neumann's theorem states that  and  are self-adjoint, and that  and  both have bounded inverses. If  has trivial kernel,  has dense range (by the above identity.) Moreover:

 is surjective if and only if there is a  such that  for all  in  (This is essentially a variant of the so-called closed range theorem.) In particular,  has closed range if and only if  has closed range. In contrast to the bounded case, it is not necessary that  since, for example, it is even possible that  does not exist. This is, however, the case if, for example,  is bounded. A densely defined, closed operator  is called normal if it satisfies the following equivalent conditions:

 ;
 the domain of  is equal to the domain of  and  for every  in this domain;
 there exist self-adjoint operators  such that  and  for every  in the domain of . Every self-adjoint operator is normal. Transpose 

Let  be an operator between Banach spaces. Then the transpose (or dual)  of  is the linear operator satisfying:

for all  and  Here, we used the notation: 

The necessary and sufficient condition for the transpose of  to exist is that  is densely defined (for essentially the same reason as to adjoints, as discussed above.) For any Hilbert space  there is the anti-linear isomorphism:

given by  where  
Through this isomorphism, the transpose  relates to the adjoint  in the following way:

where . (For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.) Note that this gives the definition of adjoint in terms of a transpose. Closed linear operators 

Closed linear operators are a class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the spectrum and  (with certain assumptions) functional calculus for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the derivative and a large class of differential operators. Let  be two Banach spaces. A linear operator  is closed if for every sequence  in  converging to  in  such that  as  one has  and . Equivalently,  is closed if its graph is closed in the direct sum . Given a linear operator , not necessarily closed, if the closure of its graph in  happens to be the graph of some operator, that operator is called the closure of , and we say that  is closable. Denote the closure of  by . It follows that  is the restriction of  to . A core (or essential domain) of a closable operator is a subset  of  such that the closure of the restriction of  to  is . Example 

Consider the derivative operator  where  is the Banach space of all continuous functions on an interval . If one takes its domain  to be , then  is a closed operator which is not bounded. On the other hand if , then  will no longer be closed, but it will be closable, with the closure being its extension defined on . Symmetric operators and self-adjoint operators 

An operator T on a Hilbert space is symmetric if and only if for each x and y in the domain of  we have . A densely defined operator  is symmetric if and only if it agrees with its adjoint T∗ restricted to the domain of T, in other words when T∗ is an extension of . In general, if T is densely defined and symmetric, the domain of the adjoint T∗ need not equal the domain of T. If T is symmetric and the domain of T and the domain of the adjoint coincide, then we say that T is self-adjoint. Note that, when T is self-adjoint, the existence of the adjoint implies that T is densely defined and since T∗ is necessarily closed, T is closed. A densely defined operator T is symmetric, if the subspace  (defined in a previous section) is orthogonal to its image  under J (where J(x,y):=(y,-x)). Equivalently, an operator T is self-adjoint if it is densely defined, closed, symmetric, and satisfies the fourth condition: both operators ,  are surjective, that is, map the domain of T onto the whole space H. In other words: for every x in H there exist y and z in the domain of T such that  and . An operator T is self-adjoint, if the two subspaces ,  are orthogonal and their sum is the whole space 

This approach does not cover non-densely defined closed operators. Non-densely defined symmetric operators can be defined directly or via graphs, but not via adjoint operators. A symmetric operator is often studied via its Cayley transform. An operator T on a complex Hilbert space is symmetric if and only if its quadratic form is real, that is, the number  is real for all x in the domain of T.

A densely defined closed symmetric operator T is self-adjoint if and only if T∗ is symmetric. It may happen that it is not. A densely defined operator T is called positive (or nonnegative) if its quadratic form is nonnegative, that is,  for all x in the domain of T. Such operator is necessarily symmetric. The operator T∗T is self-adjoint and positive for every densely defined, closed T.

The spectral theorem applies to self-adjoint operators  and moreover, to normal operators, but not to densely defined, closed operators in general, since in this case the spectrum can be empty. A symmetric operator defined everywhere is closed, therefore bounded, which is the Hellinger–Toeplitz theorem. Extension-related

By definition, an operator T is an extension of an operator S if . An equivalent direct definition: for every x in the domain of S, x belongs to the domain of T and . Note that an everywhere defined extension exists for every operator, which is a purely algebraic fact explained at  and based on the axiom of choice. If the given operator is not bounded then the extension is a discontinuous linear map. It is of little use since it cannot preserve important properties of the given operator (see below), and usually is highly non-unique. An operator T is called closable if it satisfies the following equivalent conditions:
 T has a closed extension;
 the closure of the graph of T is the graph of some operator;
 for every sequence (xn) of points from the domain of T such that xn → 0 and also Txn → y it holds that . Not all operators are closable. A closable operator T has the least closed extension  called the closure of T. The closure of the graph of T is equal to the graph of  Other, non-minimal closed extensions may exist. A densely defined operator T is closable if and only if T∗ is densely defined. In this case  and 

If S is densely defined and T is an extension of S then S∗ is an extension of T∗. Every symmetric operator is closable. A symmetric operator is called maximal symmetric if it has no symmetric extensions, except for itself. Every self-adjoint operator is maximal symmetric. The converse is wrong. An operator is called essentially self-adjoint if its closure is self-adjoint. An operator is essentially self-adjoint if and only if it has one and only one self-adjoint extension. A symmetric operator may have more than one self-adjoint extension, and even a continuum of them. A densely defined, symmetric operator T is essentially self-adjoint if and only if both operators ,  have dense range. Let T be a densely defined operator. Denoting the relation "T is an extension of S" by S ⊂ T (a conventional abbreviation for Γ(S) ⊆ Γ(T)) one has the following. If T is symmetric then T ⊂ T∗∗ ⊂ T∗. If T is closed and symmetric then T = T∗∗ ⊂ T∗. If T is self-adjoint then T = T∗∗ = T∗. If T is essentially self-adjoint then T ⊂ T∗∗ = T∗. Importance of self-adjoint operators
The class of self-adjoint operators is especially important in mathematical physics. Every self-adjoint operator is densely defined, closed and symmetric. The converse holds for bounded operators but fails in general. Self-adjointness is substantially more restricting than these three properties. The famous spectral theorem holds for self-adjoint operators. In combination with Stone's theorem on one-parameter unitary groups it shows that self-adjoint operators are precisely the infinitesimal generators of strongly continuous one-parameter unitary groups, see . Such unitary groups are especially important for describing time evolution in classical and quantum mechanics. See also 
 
 Stone–von Neumann theorem
 Bounded operator

Notes

References

Citations

Bibliography 

  (see Chapter 12 "General theory of unbounded operators in Hilbert spaces"). (see Chapter 5 "Unbounded operators"). (see Chapter 8 "Unbounded operators"). Linear operators
Operator theory

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