unbounded operator

Some of the most important operators in analysis and physics — differentiation, the position and momentum operators, the Laplacian — are unbounded: they can amplify a unit-length input into an arbitrarily large output. Such operators cannot act on the whole space without exploding, so they live only on a carefully chosen dense subspace, their domain.

An unbounded operator T on a Hilbert space H is a linear map defined on a dense subspace D(T) of H (its domain) for which no finite constant C makes ||T x|| <= C ||x|| hold for all x in D(T). The domain is part of the data: the very same formula on different domains is a different operator. Densely defined is required so that the adjoint can be defined at all.

Continuity is lost — an unbounded operator is necessarily discontinuous — so the convenient theory of bounded operators does not apply verbatim. Instead one demands the operator be closed (its graph is closed in H x H), which is the right substitute that keeps spectral theory and the closed graph perspective intact. The closed graph theorem warns that a closed operator defined on ALL of H would automatically be bounded, which is exactly why genuine unbounded operators must have proper dense domains.