Borel Functional Calculus - The Bounded Functional Calculus

The Bounded Functional Calculus

Formally, the bounded Borel functional calculus of a self adjoint operator T on Hilbert space H is a mapping defined on the space of bounded complex-valued Borel functions f on the real line,

such that the following conditions hold

• π_T is an involution preserving and unit-preserving homomorphism from the ring of complex-valued bounded measurable functions on R.

• If ξ is an element of H, then

is a countably additive measure on the Borel sets of R. In the above formula 1_E denotes the indicator function of E. These measures ν_ξ are called the spectral measures of T.

where η denotes the mapping z → z on C.

Theorem. Any self-adjoint operator T has a unique Borel functional calculus.

This defines the functional calculus for bounded functions applied to possibly unbounded self-adjoint operators. Using the bounded functional calculus, one can prove part of the Stone's theorem on one-parameter unitary groups:

Theorem. If A is a self-adjoint operator, then

is a 1-parameter strongly continuous unitary group whose infinitesimal generator is i A.

As an application, we consider the Schrödinger equation, or equivalently, the dynamics of a quantum mechanical system. In non-relativistic quantum mechanics, the Hamiltonian operator H models the total energy observable of a quantum mechanical system S. The unitary group generated by i H corresponds to the time evolution of S.

We can also use the Borel functional calculus to abstractly solve some linear initial value problems such as the heat equation, or Maxwell's equations.