closed operator (original) (raw)

Given an operator A, not necessarily closed, if the closure of its graph in B⊕B 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:

2. 1. If A is closed then A-λ⁢I is closed;
3. 1. If A is closed and it has an inverse, then A-1 is also closed;
4. 1. An operator A admits a closure if and only if for every pair of sequences {xn} and {yn} in 𝒟⁢(A), both converging to z∈B, and such that both {A⁢xn} and {A⁢yn} converge, it holds limn⁡A⁢xn=limn⁡A⁢yn.