Closability

Closability is a property of linear operators between topological vector spaces, typically between Banach spaces in functional analysis. It concerns whether an operator that is defined on a limited domain can be meaningfully extended to a larger, closed operator using limit processes.

Let T be a linear operator from a domain D(T) in X to Y. T is closable if the closure of its graph, the set of pairs (x, T x) with x in D(T), is the graph of some operator. Equivalently, T is closable if, whenever a sequence x_n in D(T) satisfies x_n → 0 in X and T x_n → y in Y, we must have y = 0. This criterion prevents nontrivial limit behavior that would prevent a well-defined extension.

If T is closable, its closure T̄ is defined on the set of all x for which there exists a sequence x_n ∈ D(T) with x_n → x and T x_n → y for some y in Y. For such x, T̄ x is defined as that limit y. The operator T̄ is closed, meaning its graph is a closed subset of X × Y. If T itself is already closed, then T̄ = T.

Closable operators arise frequently in analysis, especially for unbounded operators. A densely defined operator is closable if and only if it has a unique closed extension, its closure. Examples include differential operators acting on appropriate function spaces, where the closure often corresponds to a weaker notion of derivative (e.g., distributional or weak derivatives, leading to Sobolev spaces).

See also: closed operator, densely defined operator, adjoint, graph of an operator.