# Closed operator

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.

${\displaystyle A\colon {\mathcal {D}}(A)\subset X\to Y}$

Given a linear operator ${\displaystyle A}$, not necessarily closed, if the closure of its graph in ${\displaystyle X\oplus Y}$ happens to be the graph of some operator, that operator is called the closure of ${\displaystyle A}$, and we say that ${\displaystyle A}$ is closable. Denote the closure of ${\displaystyle A}$ by ${\displaystyle {\overline {A}}.}$ It follows easily that ${\displaystyle A}$ is the restriction of ${\displaystyle {\overline {A}}}$ to ${\displaystyle {\mathcal {D}}(A).}$

## Basic Properties

The following properties are easily checked:

## Example

Consider the derivative operator

${\displaystyle Af=f'\,}$

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 ${\displaystyle {\mathcal {D}}(A)}$ to be ${\displaystyle {\mathcal {D}}(A)=C^{1}[a,b]}$, then A is a closed operator, which is not bounded. (Note that one could also set ${\displaystyle {\mathcal {D}}(A)}$ to be the set of all differentiable functions including those with non-continuous derivative. That operator is not closed!)

If one takes ${\displaystyle {\mathcal {D}}(A)}$ 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 ${\displaystyle C^{1}[a,b]}$.