Closed linear operator explained

In functional analysis, a branch of mathematics, a closed linear operator or often a closed operator is a linear operator whose graph is closed (see closed graph property). It is a basic example of an unbounded operator.

The closed graph theorem says a linear operator between Banach spaces is a closed operator if and only if it is a bounded operator. Hence, a closed linear operator that is used in practice is typically only defined on defined on a dense subspace of a Banach space.

Definition

It is common in functional analysis to consider partial functions, which are functions defined on a subset of some space

X. A partial function f is declared with the notation f : D \subseteq X \to Y, which indicates that f has prototype f : D \to Y (that is, its domain is D and its codomain is Y )

Every partial function is, in particular, a function and so all terminology for functions can be applied to them. For instance, the graph of a partial function

f is the set $\backslash operatorname\; =\; \backslash .$ However, one exception to this is the definition of "closed graph". A function f : D \subseteq X \to Y is said to have a closed graph if \operatorname{graph} f is a closed subset of X x Y in the product topology; importantly, note that the product space is X x Y and D x Y = \operatorname{dom} f x Y as it was defined above for ordinary functions. In contrast, when f : D \to Y is considered as an ordinary function (rather than as the partial function f : D \subseteq X \to Y ), then "having a closed graph" would instead mean that \operatorname{graph} f is a closed subset of D x Y. If \operatorname{graph} f is a closed subset of X x Y then it is also a closed subset of \operatorname{dom} (f) x Y although the converse is not guaranteed in general.

Definition: If and are topological vector spaces (TVSs) then we call a linear map a closed linear operator if its graph is closed in .

Closable maps and closures

A linear operator

f : D \subseteq X \to Y is in X x Y if there exists a E \subseteq X containing D and a function (resp. multifunction) F : E \to Y whose graph is equal to the closure of the set \operatorname{graph} f in X x Y. Such an F is called a closure of f in X x Y , is denoted by \overline{f}, and necessarily extends f.

If

f : D \subseteq X \to Y is a closable linear operator then a or an of f is a subset C \subseteq D such that the closure in X x Y of the graph of the restriction f \ vert _{C} : C \to Y of f to C is equal to the closure of the graph of f in X x Y (i.e. the closure of \operatorname{graph} f in X x Y is equal to the closure of \operatorname{graph} f \ vert _{C} in X x Y ).

Examples

A bounded operator is a closed operator. Here are examples of closed operators that are not bounded.

• If
  (X, \tau) is a Hausdorff TVS and \nu is a vector topology on X that is strictly finer than \tau, then the identity map \operatorname{Id} : (X, \tau) \to (X, \nu) a closed discontinuous linear operator.
• Consider the derivative operator A = (d)/(d x) where X = Y = C([a, b]). is the Banach space of all continuous functions on an interval [a, b]. If one takes its domain D(f) to be C ^{1([a,} b]), then f is a closed operator, which is not bounded.^[1] On the other hand, if D(f) is the space C ^{infty([a,} b]) of smooth functions scalar valued functions then f will no longer be closed, but it will be closable, with the closure being its extension defined on C ^{1([a,} b]).

Basic properties

The following properties are easily checked for a linear operator between Banach spaces:

• If is closed then is closed where is a scalar and is the identity function;
• If is closed, then its kernel (or nullspace) is a closed vector subspace of ;
• If is closed and injective then its inverse is also closed;
• A linear operator admits a closure if and only if for every and every pair of sequences and in both converging to in, such that both and converge in, one has .

Notes and References

1. Book: Kreyszig, Erwin. Introductory Functional Analysis With Applications. John Wiley & Sons. Inc.. 1978. 0-471-50731-8. USA. 294.