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 (xn) of points from the domain of T such that xn converge to some x and Txn 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.
Read more about this topic: Closed Operator
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