Closed Operator - Definitions and Basic Properties

Definitions and Basic Properties

Let B₁ and B₂ be Banach spaces. An unbounded operator (or simply operator) T : B₁ → B₂ is a linear map T from a linear subspace D(T) of B₁ — the domain of T — to the space B₂. Contrary to the usual convention, T may not be defined on the whole space B₁. Two operators are equal if they have the common domain and they coincide on that common domain.

An operator T is said to be closed if its graph Γ(T) is a closed set. (Here, the graph Γ(T) is a linear subspace of the direct sum B₁ ⊕ B₂, defined as the set of all pairs (x, Tx), where x runs over the domain of T). Explicitly, this means that for every sequence (x_n) of points from the domain of T such that x_n converge to some x and Tx_n converge to some y, it holds that x belongs to the domain of T and Tx = y. The closedness can also be formulated in terms of the graph norm: an operator T is closed if and only if its domain D(T) is a complete space with respect to the norm:

An operator T is said to be densely defined if its domain is dense in B₁. This also includes operators defined on the entire space B₁, since the whole space is dense in itself. The denseness of the domain is necessary and sufficient for the existence of the adjoint and the transpose; see the next section.

If T : B₁ → B₂ is closed, densely defined and continuous on its domain, then it is defined on B₁.

A densely defined operator T on a Hilbert space H is called bounded from below if T + a is a positive operator for some real number a. That is, ⟨Tx|x⟩ ≥ −a·||x||2 for all x in the domain of T. If both T and (–T) are bounded from below then T is bounded.