Closed operator

An operator $ A : D _ {A} \rightarrow Y $ such that if $ x _ {n} \in D _ {A} $, $ x _ {n} \rightarrow x $ and $ A x _ {n} \rightarrow y $, then $ x \in D _ {A} $ and $ A x = y $. (Here $ X , Y $ are Banach spaces over the same field of scalars and $ D \subset X $ is the domain of definition of $ A $.) The notion of a closed operator may be extended to operators defined on separable linear topological spaces, except that instead of a sequence $ \{ x _ {n} \} $ one must consider arbitrary directions (nets) $ \{ x _ \xi \} $. If $ \mathop{\rm Gr} A $ is the graph of $ A $, then $ A $ is closed if and only if $ \mathop{\rm Gr} A $ is a closed subset of the Cartesian product $ X \times Y $. 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 $ A = d / dt $, defined on the set $ C _ {1} [ a , b ] $ of continuously-differentiable functions in the space $ C [ a , b ] $. Many operators of mathematical physics are closed but not continuous.

An operator $ A $ 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 $ x _ {n} , x _ {n} ^ \prime \in D _ {A} $,

$$ \lim\limits x _ {n} = \lim\limits x _ {n} ^ \prime ,\ \lim\limits A x _ {n} = y ,\ \lim\limits A x _ {n} ^ \prime = y ^ \prime , $$

that $ y = y ^ \prime $. 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 $ A : X \rightarrow Y $ is closed. Conversely, if $ A $ is defined on all of $ X $ and closed, then it is bounded. If $ A $ is closed and $ A ^ {- 1} $ exists, then $ A ^ {- 1} $ is also closed. Since $ A : X \rightarrow X $ is closed if and only if $ A - \lambda I $ is closed, it follows that $ A $ is closed if the resolvent $ R _ \lambda ( A ) = ( A - \lambda I ) ^ {- 1} $ exists and is bounded for at least one value of the parameter $ \lambda \in \mathbf C $.

If $ D _ {A} $ is dense in $ X $ and, consequently, the adjoint operator $ A ^ {*} : D _ {A ^ {*} } \rightarrow X ^ {*} $, $ D _ {A ^ {*} } \subset Y ^ {*} $, is uniquely defined, then $ A ^ {*} $ is a closed operator. If, moreover, $ D _ {A ^ {*} } $ is dense in $ Y ^ {*} $ and $ X , Y $ are reflexive, then $ A $ is a closeable operator and its closure is $ A ^ {**} $.

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

$$ \| x \| _ {0} = \| x \| _ {X} + \| Ax \| _ {Y} . $$

Then $ D _ {A} $ with this new norm is a Banach space and $ A $, as an operator from $ ( D _ {A} , \| \cdot \| _ {0} ) $ to $ Y $, 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)