Closed Linear Operators
Closed linear operators are a 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.
Let be two Banach spaces. A linear operator
is closed if for every sequence in converging to such that as one has and Equivalently, is closed if its graph is closed in the direct sum
Given a linear operator, not necessarily closed, if the closure of its graph in happens to be the graph of some operator, that operator is called the closure of, and we say that is closable. 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
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