closed operator

Let B be a Banach spaceMathworldPlanetmath. A linear operatorMathworldPlanetmath A:𝒟(A)BB is said to be if for every sequence {xn}n in 𝒟(A) converging to xB such that AxnnyB, it holds x𝒟(A) and Ax=y. Equivalently, A is closed if its graph is closed in BB.

Given an operator A, not necessarily closed, if the closure of its graph in BB happens to be the graph of some operator, we call that operator the closure of A, and we say that A is closable. We denote the closure of A by A¯. It follows easily that A is the restriction of A¯ to 𝒟(A).

A core of a closable operator is a subset 𝒞 of 𝒟(A) such that the closure of the restriction of A to 𝒞 is A¯.

The following properties are easily checked:

  1. 1.

    Any bounded linear operator defined on the whole space B is closed;

  2. 2.

    If A is closed then A-λI is closed;

  3. 3.

    If A is closed and it has an inverse, then A-1 is also closed;

  4. 4.

    An operator A admits a closure if and only if for every pair of sequences {xn} and {yn} in 𝒟(A), both converging to zB, and such that both {Axn} and {Ayn} converge, it holds limnAxn=limnAyn.

Title closed operator
Canonical name ClosedOperator
Date of creation 2013-03-22 13:48:20
Last modified on 2013-03-22 13:48:20
Owner Koro (127)
Last modified by Koro (127)
Numerical id 9
Author Koro (127)
Entry type Definition
Classification msc 47A05
Synonym closed
Defines closure
Defines closable
Defines core