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.

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 .

: <math>\|x\|_T = \sqrt{ \|x\|^2 + \|Tx\|^2 }.</math>

An operator is said to be densely defined if its domain is dense in .

A densely defined symmetric 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). It may happen that the domain of <math>T^*</math> is a closed hyperplane and <math>T^*</math> vanishes everywhere on the domain. Thus, boundedness of <math>T^*</math> on its domain does not imply boundedness of . On the other hand, if <math>T^*</math> 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 <math>T^*</math> is dense, then it has its adjoint <math>T^{**}.</math>

The other equivalent definition of the adjoint can be obtained by noticing a general fact. Define a linear operator <math>J</math> as follows:

<math display=block>\begin{cases} J: H_1 \oplus H_2 \to H_2 \oplus H_1 \\ J(x \oplus y) = -y \oplus x \end{cases}</math>

Since <math>J</math> is an isometric surjection, it is unitary. Hence: <math>J(\Gamma(T))^{\bot}</math> is the graph of some operator <math>S</math> if and only if is densely defined. A simple calculation shows that this "some" <math>S</math> satisfies:

<math display=block>\langle Tx \mid y \rangle_2 = \langle x \mid Sy \rangle_1,</math>

for every in the domain of . Thus <math>S</math> is the adjoint of .

It follows immediately from the above definition that the adjoint <math>T^*</math> is closed.

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 <math>T : H_1 \to H_2</math> coincides with the orthogonal complement of the range of the adjoint. That is,

<math display=block>\operatorname{ker}(T) = \operatorname{ran}(T^*)^\bot.</math>

von Neumann's theorem states that <math>T^* T</math> and <math>T T^*</math> are self-adjoint, and that <math>I + T^* T</math> and <math>I + T T^*</math> both have bounded inverses. If <math>T^*</math> has trivial kernel, has dense range (by the above identity.) Moreover:

: is surjective if and only if there is a <math>K > 0</math> such that <math>\|f\|_2 \leq K \left\|T^* f\right\|_1</math> for all <math>f</math> in <math>D\left(T^*\right).</math> (This is essentially a variant of the so-called closed range theorem.) In particular, has closed range if and only if <math>T^*</math> has closed range.

In contrast to the bounded case, it is not necessary that <math>(T S)^* = S^* T^*,</math> since, for example, it is even possible that <math>(T S)^*</math> 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:

<math>T^* T = T T^*</math>;
the domain of is equal to the domain of <math>T^*,</math> and <math>\|T x\| = \left\|T^* x\right\|</math> for every in this domain;
there exist self-adjoint operators <math>A, B</math> such that <math>T = A + i B,</math><math>T^* = A - i B,</math> and <math>\|T x\|^2 = \|A x\|^2 + \|B x\|^2</math> for every in the domain of .

Every self-adjoint operator is normal.

Transpose

Let <math>T : B_1 \to B_2</math> be an operator between Banach spaces. Then the transpose (or dual) <math>{}^t T: {B_2}^* \to {B_1}^*</math> of <math>T</math> is the linear operator satisfying:

<math display=block>\langle T x, y' \rangle = \langle x, \left({}^t T\right) y' \rangle</math>

for all <math>x \in B_1</math> and <math>y \in B_2^*.</math> Here, we used the notation: <math>\langle x, x' \rangle = x'(x).</math>

The necessary and sufficient condition for the transpose of <math>T</math> to exist is that <math>T</math> is densely defined (for essentially the same reason as to adjoints, as discussed above.)

For any Hilbert space <math>H,</math> there is the anti-linear isomorphism:

given by <math>J f = y</math> where <math>f(x) = \langle x \mid y \rangle_H, (x \in H).</math>

Through this isomorphism, the transpose <math>{}^t T</math> relates to the adjoint <math>T^*</math> in the following way:

<math display=block>T^* = J_1 \left({}^t T\right) J_2^{-1},</math>

where <math>J_j: H_j^* \to H_j</math>. (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 <math>\langle Tx \mid y \rangle = \lang x \mid Ty \rang</math>. 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 <math> H \oplus H .</math> 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, <math>\langle Tx \mid x \rangle \ge 0 </math> for all x in the domain of T. Such operator is necessarily symmetric.

The operator T∗T is self-adjoint and positive 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,

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 .

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 <math> \overline T </math> called the closure of T. The closure of the graph of T is equal to the graph of <math> \overline T. </math>

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.

An operator is called essentially self-adjoint if its closure is self-adjoint.

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.

Unbounded operator

Short history

Definitions and basic properties

Transpose

Closed linear operators

Example

Symmetric operators and self-adjoint operators

Importance of self-adjoint operators

See also

Notes

References

Citations

Bibliography

Short history

Definitions and basic properties

Transpose

Closed linear operators

Example

Symmetric operators and self-adjoint operators

Extension-related

Importance of self-adjoint operators

See also

Notes

References

Citations

Bibliography