Self-adjoint Operator - Extensions of Symmetric Operators

Extensions of Symmetric Operators

Further information: Extensions of symmetric operators and unbounded operator

The following question arises in several contexts: if an operator A on the Hilbert space H is symmetric, when does it have self-adjoint extensions? One answer is provided by the Cayley transform of a self-adjoint operator and the deficiency indices. (We should note here that it is often of technical convenience to deal with closed operators. In the symmetric case, the closedness requirement poses no obstacles, since it is known that all symmetric operators are closable.)

Theorem. Suppose A is a symmetric operator. Then there is a unique partially defined linear operator

such that

Here, ran and dom denote the range and the domain, respectively. W(A) is isometric on its domain. Moreover, the range of 1 − W(A) is dense in H.

Conversely, given any partially defined operator U which is isometric on its domain (which is not necessarily closed) and such that 1 − U is dense, there is a (unique) operator S(U)

such that

The operator S(U) is densely defined and symmetric.

The mappings W and S are inverses of each other.

The mapping W is called the Cayley transform. It associates a partially defined isometry to any symmetric densely defined operator. Note that the mappings W and S are monotone: This means that if B is a symmetric operator that extends the densely defined symmetric operator A, then W(B) extends W(A), and similarly for S.

Theorem. A necessary and sufficient condition for A to be self-adjoint is that its Cayley transform W(A) be unitary.

This immediately gives us a necessary and sufficient condition for A to have a self-adjoint extension, as follows:

Theorem. A necessary and sufficient condition for A to have a self-adjoint extension is that W(A) have a unitary extension.

A partially defined isometric operator V on a Hilbert space H has a unique isometric extension to the norm closure of dom(V). A partially defined isometric operator with closed domain is called a partial isometry.

Given a partial isometry V, the deficiency indices of V are defined as the dimension of the orthogonal complements of the domain and range:

Theorem. A partial isometry V has a unitary extension if and only if the deficiency indices are identical. Moreover, V has a unique unitary extension if and only if the both deficiency indices are zero.

We see that there is a bijection between symmetric extensions of an operator and isometric extensions of its Cayley transform. An operator which has a unique self-adjoint extension is said to be essentially self-adjoint. Such operators have a well-defined Borel functional calculus. Symmetric operators which are not essentially self-adjoint may still have a canonical self-adjoint extension. Such is the case for non-negative symmetric operators (or more generally, operators which are bounded below). These operators always have a canonically defined Friedrichs extension and for these operators we can define a canonical functional calculus. Many operators that occur in analysis are bounded below (such as the negative of the Laplacian operator), so the issue of essential adjointness for these operators is less critical.

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