Closed Operator

Adjoint

The adjoint of an unbounded operator can be defined in two equivalent ways. First, it can be defined in a way analogous to how we define the adjoint of a bounded operator. Namely, the adjoint T∗ : H₂ → H₁ of T is defined as an operator with the property:

More precisely, T∗ is defined in the following way. If y is such that is a continuous linear functional on the domain of T, then, after extending it to the whole space via the Hahn–Banach theorem, we can find a z such that

since the dual of a Hilbert space can be identified with the set of linear functionals given by the inner product. For each y, z is uniquely determined if and only if the linear functional is densely defined; i.e., T is densely defined. Finally, we let T∗y = z, completing the construction of T∗. Note that T∗ exists if and only if T is densely defined.

By definition, the domain of T∗ consists of elements such that is continuous on the domain of T. Consequently, the domain of T∗ could be anything; it could be trivial (i.e., contains only zero) It may happen that the domain of T∗ is a closed hyperplane and T∗ vanishes everywhere on the domain. Thus, boundedness of T∗ on its domain does not imply boundedness of T. On the other hand, if T∗ is defined on the whole space then T is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space. If the domain of T∗ is dense, then it has its adjoint T∗∗. A closed densely defined operator T is bounded if and only if T∗ is bounded.

The other equivalent definition of the adjoint can be obtained by noticing a general fact: define a linear operator by . (Since is an isometric surjection, it is unitary.) We then have: is the graph of some operator S if and only if is densely defined. A simple calculation shows that this "some" S satisfies: for every x in the domain of T. Thus, S is the adjoint of T.

It follows immediately from the above definition that the adjoint T∗ is closed. In particular, a self-adjoint operator (i.e., T = T∗) is closed. An operator T is closed and densely defined if and only if T∗∗ = T.

Some well-known properties for bounded operators generalize to closed densely defined operators. The kernel of a closed operator is closed. Moreover, the kernel of a closed densely defined operator T : H₁ → H₂ coincides with the orthogonal complement of the range of the adjoint. That is,

von Neumann's theorem states that T∗T and TT∗ are self-adjoint, and that I + T∗T and I + TT∗ both have bounded inverses. If has trivial kernel, has dense range (by the above identity.) Moreover, T is surjective if and only if there is a such that

for every .

(This is essentially a variant of the so-called closed range theorem.) In particular, T has closed range if and only if T∗ has closed range.

In contrast to the bounded case, it is not necessary that we have: (TS)∗ = S∗T∗, since, for example, it is even possible that (TS)∗ doesn't exist. This is, however, the case if, for example, T is bounded.

A densely defined, closed operator T is called normal if it satisfies the following equivalent conditions:

T∗T = T T∗;
the domain of T is equal to the domain of T∗, and for every x in this domain;
there exist self-adjoint operators A, B such that T = A + iB, T∗ = A – iB, and for every x in the domain of T.