Closed operator

An operator such that if , and , then and . (Here are Banach spaces over the same field of scalars and is the domain of definition of .) The notion of a closed operator may be extended to operators defined on separable linear topological spaces, except that instead of a sequence one must consider arbitrary directions (nets) . If is the graph of , then is closed if and only if is a closed subset of the Cartesian product . This property is often adopted as the definition of a closed operator.

The notion of a closed operator is a generalization of the notion of an operator defined and continuous on a closed subset of a Banach space. An example of a closed but not continuous operator is , defined on the set of continuously-differentiable functions in the space . Many operators of mathematical physics are closed but not continuous.

An operator has a closure (i.e. is closeable) if it admits a closed extension. An operator has a closure if and only if it follows from ,

that . The smallest closed extension of an operator is called its closure. A symmetric operator on a Hilbert space with dense domain of definition always admits a closure.

A bounded linear operator is closed. Conversely, if is defined on all of and closed, then it is bounded. If is closed and exists, then is also closed. Since is closed if and only if is closed, it follows that is closed if the resolvent exists and is bounded for at least one value of the parameter .

If is dense in and, consequently, the adjoint operator , , is uniquely defined, then is a closed operator. If, moreover, is dense in and are reflexive, then is a closeable operator and its closure is .

A closed operator can be made bounded by introducing a new norm on its domain of definition. Let

Then with this new norm is a Banach space and , as an operator from to , is bounded.

References

[1]	K. Yosida, "Functional analysis" , Springer (1980)
[2]	T. Kato, "Perturbation theory for linear operators" , Springer (1980)

Comments

The result that a closed linear operator mapping (all of) a Banach space into a Banach space is continuous is known as the closed-graph theorem.

References

[a1]	S. Goldberg, "Unbounded linear operators" , McGraw-Hill (1966)