Borel functional calculus

If T is a self-adjoint operator on a finite-dimensional inner product space H, then H has an orthonormal basis {e₁, ..., e_ℓ} consisting of eigenvectors of T, that is

${\displaystyle Te_{k}=\lambda _{k}e_{k},\qquad 1\leq k\leq \ell .}$

Thus, for any positive integer n,

${\displaystyle T^{n}e_{k}=\lambda _{k}^{n}e_{k}.}$

If only polynomials in T are considered, then one arrives at the holomorphic functional calculus. Are more general functions of T possible? Yes. Given a Borel function h, one can define an operator h(T) by specifying its behavior on the basis:

${\displaystyle h(T)e_{k}=h(\lambda _{k})e_{k}.}$

In general, any self-adjoint operator T is unitarily equivalent to a multiplication operator; this means that for many purposes, T can be considered as an operator

${\displaystyle [T\psi ](x)=f(x)\psi (x)}$

acting on L² of some measure space. The domain of T consists of those functions for which the above expression is in L². In this case, one can define analogously

${\displaystyle [h(T)\psi ](x)=[h\circ f](x)\psi (x).}$

For many technical purposes, the preceding formulation is good enough. However, it is desirable to formulate the functional calculus in a way in which it is clear that it does not depend on the particular representation of T as a multiplication operator. This we do in the next section.

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,

${\displaystyle {\begin{cases}\pi _{T}:L^{\infty }(\mathbb {R} ,\mathbb {C} )\to {\mathcal {B}}({\mathcal {H}})\\f\mapsto f(T)\end{cases}}}$

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

${\displaystyle \nu _{\xi }:E\mapsto \langle \pi _{T}(\mathbf {1} _{E})\xi ,\xi \rangle }$

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.

• If η denotes the mapping z → z on C, then:

${\displaystyle \pi _{T}\left([\eta +i]^{-1}\right)=[T+i]^{-1}.}$

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

${\displaystyle U_{t}=e^{itA},\qquad t\in \mathbb {R} }$

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

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 iH 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.

We can also define the functional calculus for not necessarily bounded Borel functions h; the result is an operator which in general fails to be bounded. Using the multiplication by a function f model of a self-adjoint operator given by the spectral theorem, this is multiplication by the composition of h with f.

Theorem. Let T be a self-adjoint operator on H, h a real-valued Borel function on R. There is a unique operator S such that

${\displaystyle \operatorname {dom} S=\left\{\xi \in H:h\in L_{\nu _{\xi }}^{2}(\mathbb {R} )\right\}}$

${\displaystyle \langle S\xi ,\xi \rangle =\int _{\mathbb {R} }h(t)\ d\nu _{\xi }(t),\quad {\mbox{for}}\quad \xi \in \operatorname {dom} S}$

The operator S of the previous theorem is denoted h(T).

More generally, a Borel functional calculus also exists for (bounded) normal operators.

Let T be a self-adjoint operator. If E is a Borel subset of R, and 1_E is the indicator function of E, then 1_E(T) is a self-adjoint projection on H. Then mapping

${\displaystyle \Omega :E\mapsto \mathbf {1} _{E}(T)}$

is a projection-valued measure called the resolution of the identity for the self adjoint operator T. The measure of R with respect to Ω is the identity operator on H. In other words, the identity operator can be expressed as the spectral integral ${\displaystyle \textstyle {I=\int 1\,d\Omega }}$. Sometimes the term "resolution of the identity" is also used to describe this representation of the identity operator as a spectral integral.

In the case of a discrete measure (in particular, when H is finite-dimensional), ${\displaystyle \textstyle {I=\int 1\,d\Omega }}$ can be written as

${\displaystyle I=\sum _{i}\left|i\right\rangle \left\langle i\right|}$

in the Dirac notation, where each ${\displaystyle |i\rangle }$ is a normalized eigenvector of T. The set ${\displaystyle \{|i\rangle \}}$ is an orthonormal basis of H.

In physics literature, using the above as heuristic, one passes to the case when the spectral measure is no longer discrete and write the resolution of identity as

${\displaystyle I=\int d|i\rangle \langle i|}$

and speak of a "continuous basis", or "continuum of basis states", ${\displaystyle \{|i\rangle \}}$ Mathematically, unless rigorous justifications are given, this expression is purely formal.