Self-adjoint Operator - Spectral Theorem

Spectral Theorem

Partially defined operators A, B on Hilbert spaces H, K are unitarily equivalent if and only if there is a unitary transformation U:H → K such that

• U maps dom A bijectively onto dom B,

A multiplication operator is defined as follows: Let be a countably additive measure space and f a real-valued measurable function on X. An operator T of the form

whose domain is the space of ψ for which the right-hand side above is in L2 is called a multiplication operator.

Theorem. Any multiplication operator is a (densely defined) self-adjoint operator. Any self-adjoint operator is unitarily equivalent to a multiplication operator.

This version of the spectral theorem for self-adjoint operators can be proved by reduction to the spectral theorem for unitary operators. This reduction uses the Cayley transform for self-adjoint operators which is defined in the next section. We might note that if T is multiplication by f, then the spectrum of T is just the essential range of f.