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)