Template:Mergeto In mathematics, specifically in functional analysis, closed linear operators are an important 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 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 easily that is the restriction of to
A core of a closable operator is a subset of such that the closure of the restriction of to is
The following properties are easily checked:
- Any closed linear operator defined on the whole space is bounded. This is the closed graph theorem;
- 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 subspace of ;
- If is closed and injective, then its inverse is also closed;
- An operator admits a closure if and only if for every pair of sequences and in both converging to , such that both and converge, one has .
Consider the derivative operator
where the Banach space X=Y is the space C[a, b] of all continuous functions on an interval [a, b]. If one takes its domain to be , then A is a closed operator, which is not bounded. (Note that one could also set to be the set of all differentiable functions including those with non-continuous derivative. That operator is not closed!)
If one takes to be instead the set of all infinitely differentiable functions, A will no longer be closed, but it will be closable, with the closure being its extension defined on .
This article incorporates material from Closed operator on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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