Closed Operator - Symmetric Operators and Self-adjoint Operators

Symmetric Operators and Self-adjoint Operators

A densely defined operator T is symmetric if for all elements x and y in the domain of T.

An operator T is said to be self-adjoint if T∗ = T. Note that, when T is self-adjoint, the existence of the adjoint implies that T is dense and since is necessarily closed, is closed.

A densely defined operator T is symmetric, if the subspace Γ ( T ) is orthogonal to its image J ( Γ ( T ) ) under J.

Equivalently, an operator T is self-adjoint if it is densely defined, closed, symmetric, and satisfies the fourth condition: both operators T – i, T + i are surjective, that is, map the domain of T onto the whole space H. In other words: for every x in H there exist y and z in the domain of T such that Ty – iy = x and Tz + iz = x.

An operator T is self-adjoint, if the two subspaces Γ ( T ), J ( Γ ( T ) ) are orthogonal and their sum is the whole space

A densely defined operator T is symmetric if T∗ is an extension of T (see below).

This approach does not cover non-densely defined closed operators. Non-densely defined symmetric operators can be defined directly or via graphs, but not via adjoint operators.

A symmetric operator is often studied via its Cayley transform.

An operator T is symmetric if and only if its quadratic form is real, that is, the number is real for all x in the domain of T.

A densely defined closed symmetric operator T is self-adjoint if and only if T∗ is symmetric. It may happen that it is not.

A densely defined operator T is called positive (or nonnegative) if its quadratic form is nonnegative, that is, for all x in the domain of T. Such operator is necessarily symmetric.

The operator T∗T is self-adjoint and positive for every densely defined, closed T.

The spectral theorem applies to self-adjoint operators and moreover, to normal operators, but not to densely defined, closed operators in general, since in this case the spectrum can be empty.

A symmetric operator defined everywhere is closed, therefore bounded, which is the Hellinger–Toeplitz theorem.