Given an operator , not necessarily closed, if the closure of its graph in happens to be the graph of some operator, we call that operator the closure of , and we say that is closable. We denote the closure of by . It follows easily that is the restriction of to .
A core of a closable operator is a subset of such that the closure of the restriction of to is .
The following properties are easily checked:
Any bounded linear operator defined on the whole space is closed;
If is closed then is closed;
If is closed and it has an inverse, then is also closed;
An operator admits a closure if and only if for every pair of sequences and in , both converging to , and such that both and converge, it holds .
|Date of creation||2013-03-22 13:48:20|
|Last modified on||2013-03-22 13:48:20|
|Last modified by||Koro (127)|