Editing Borel functional calculus (section)

== 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,
<math display="block">\begin{cases} \pi_T: L^\infty(\mathbb{R},\mathbb{C}) \to \mathcal{B}(\mathcal{H})\\ f \mapsto f(T) \end{cases}</math>
such that the following conditions hold

* {{mvar|π<sub>T</sub>}} is an [[involution (mathematics)|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 <math display="block"> \nu_\xi:E \mapsto \langle \pi_T(\mathbf{1}_E) \xi, \xi \rangle </math> is a [[countably additive measure]] on the Borel sets ''E'' of '''R'''. In the above formula '''1'''<sub>''E''</sub> denotes the [[indicator function]] of ''E''.  These measures ν<sub>ξ</sub> are called the '''spectral measures''' of ''T''.
* If {{mvar|η}} denotes the mapping ''z'' → ''z'' on '''C''', then: <math display="block"> \pi_T \left ([\eta +i]^{-1} \right ) = [T + i]^{-1}.</math>

{{math 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]]:

{{math theorem | If ''A'' is a self-adjoint operator, then
<math display="block"> U_t = e^{i t A}, \qquad t \in \mathbb{R} </math>
is a 1-parameter strongly continuous unitary group whose [[Lie group#The Lie algebra associated with a Lie group|infinitesimal generator]] is ''iA''.}}

As an application, we consider the [[Schrödinger equation]], or equivalently, the [[Dynamics (mechanics)|dynamics]] of a quantum mechanical system. In [[theory of relativity|non-relativistic]] [[quantum mechanics]], the [[Hamiltonian (quantum mechanics)|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 problem]]s such as the heat equation, or Maxwell's equations.

=== Existence of a functional calculus ===
The existence of a mapping with the properties of a functional calculus requires proof. For the case of a bounded self-adjoint operator ''T'', the existence of a Borel functional calculus can be shown in an elementary way as follows:

First pass from polynomial to [[continuous functional calculus]] by using the [[Stone–Weierstrass theorem]]. The crucial fact here is that, for a bounded self adjoint operator ''T'' and a polynomial ''p'',
<math display="block">\| p(T) \| = \sup_{\lambda \in \sigma(T)} |p(\lambda)|.</math>

Consequently, the mapping
<math display="block"> p \mapsto p(T) </math>
is an isometry and a densely defined homomorphism on the ring of polynomial functions. Extending by continuity defines ''f''(''T'') for a continuous function ''f'' on the spectrum of ''T''. The [[Riesz-Markov theorem]] then allows us to pass from integration on continuous functions to [[spectral measure]]s, and this is the Borel functional calculus.

Alternatively, the continuous calculus can be obtained via the [[Gelfand transform]], in the context of commutative Banach algebras. Extending to measurable functions is achieved by applying Riesz-Markov, as above. In this formulation, ''T'' can be a [[normal operator]].

Given an operator ''T'', the range of the continuous functional calculus ''h'' → ''h''(''T'') is the (abelian) C*-algebra ''C''(''T'') generated by ''T''. The Borel functional calculus has a larger range,  that is the closure of ''C''(''T'') in the [[weak operator topology]], a (still abelian) [[von Neumann algebra]].