Editing Borel functional calculus

{{Short description|Branch of functional analysis}}
{{More citations needed|date=October 2024}}
In [[functional analysis]], a branch of [[mathematics]], the '''Borel functional calculus''' is a ''[[functional calculus]]'' (that is, an assignment of [[operator (mathematics)|operators]] from [[commutative algebra]]s to functions defined on their [[Spectrum of a ring|spectra]]), which has particularly broad scope.<ref>{{cite book| isbn = 0-8218-0819-2 | title = Fundamentals of the Theory of Operator Algebras: Vol 1 | year = 1997 | publisher = Amer Mathematical Society| last1 = Kadison | first1 = Richard V. | last2 = Ringrose | first2 = John R.}}</ref><ref>{{cite book |isbn = 0-12-585050-6 | title = Methods of Modern Mathematical Physics | year = 1981 | publisher = Academic Press | first1 = Michael | last1 = Reed | first2 = Barry | last2 = Simon}}</ref> Thus for instance if ''T'' is an operator, applying the squaring function ''s'' → ''s''<sup>2</sup> to ''T'' yields the operator ''T''<sup>2</sup>. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) [[Laplacian operator]] {{math|−Δ}} or the exponential <math> e^{it \Delta}.</math>

The 'scope' here means the kind of ''function of an operator'' which is allowed. The Borel functional calculus is more general than the [[continuous functional calculus]], and its focus is different than the [[holomorphic functional calculus]].

More precisely, the Borel functional calculus allows for applying an arbitrary [[Borel function]] to a [[self-adjoint operator]], in a way that generalizes applying a [[polynomial function]].

== Motivation ==
If ''T'' is a self-adjoint operator on a finite-dimensional [[inner product space]] ''H'', then ''H'' has an [[orthonormal basis]] {{math|{''e''<sub>1</sub>, ..., ''e<sub>ℓ</sub>''} }} consisting of [[eigenvector]]s of ''T'', that is
<math display="block"> T e_k = \lambda_k e_k, \qquad 1 \leq k \leq \ell.</math>

Thus, for any positive integer ''n'',
<math display="block"> T^n e_k = \lambda_k^n e_k.</math>

If only polynomials in ''T'' are considered, then one gets the [[holomorphic functional calculus]]. The relation also holds for more general functions of ''T''. Given a [[Borel function]] ''h'', one can define an operator ''h''(''T'') by specifying its behavior on the basis:
<math display="block"> h(T) e_k = h(\lambda_k) e_k.</math>

Generally, any self-adjoint operator ''T'' is [[Self-adjoint operator|unitarily equivalent]] to a multiplication operator; this means that for many purposes, ''T'' can be considered as an operator
<math display="block"> [T \psi](x) = f(x) \psi(x)</math>
acting on ''L''<sup>2</sup> of some [[measure space]]. The domain of ''T'' consists of those functions whose above expression is in ''L''<sup>2</sup>. In such a case, one can define analogously
<math display="block"> [h(T) \psi](x) = [h \circ f](x) \psi(x). </math>

For many technical purposes, the previous formulation is good enough. However, it is desirable to formulate the functional calculus in a way that does not depend on the particular representation of ''T'' as a multiplication operator. That's what we do in the next 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]].

== The general functional calculus ==
{{see also|Operational calculus}}
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''.

{{math 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
<math display="block">\operatorname{dom} S = \left\{\xi \in H: h \in L^2_{\nu_\xi}(\mathbb{R}) \right\}</math>
<math display="block">\langle S \xi, \xi \rangle = \int_{\mathbb{R}} h(t) \ d\nu_{\xi} (t), \quad \text{for} \quad \xi \in \operatorname{dom} S</math>}}

The operator ''S'' of the previous theorem is denoted ''h''(''T'').

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

== Resolution of the identity ==
Let <math>T</math> be a self-adjoint operator. If <math>E</math> is a Borel subset of '''R''', and <math> \mathbf{1}_E </math> is the [[indicator function]] of ''E'', then <math> \mathbf{1}_E(T) </math> is a self-adjoint projection on ''H''.  Then mapping
<math display="block"> \Omega_T: E \mapsto \mathbf{1}_E(T)</math>
is a [[projection-valued measure]]. The measure of '''R''' with respect to <math display="inline">\Omega_T</math> is the identity operator on ''H''. In other words, the identity operator can be expressed as the spectral integral
:<math>I = \Omega_T([-\infty,\infty]) = \int_{-\infty}^{\infty} d\Omega_T</math>.

Stone's formula<ref>{{cite journal |last1=Takhtajan |first1=Leon A. |title=Etudes of the resolvent |journal=Russian Mathematical Surveys |date=2020 |volume=75 |issue=1 |pages=147–186 |doi=10.1070/RM9917 |url=https://arxiv.org/abs/2004.11950v1|arxiv=2004.11950 }}</ref> expresses the spectral measure <math>\Omega_T</math> in terms of the [[Resolvent formalism|resolvent]] <math>R_T(\lambda) \equiv \left(T-\lambda I \right)^{-1}</math>:
:<math>\frac{1}{2\pi i} \lim_{\epsilon \to 0^+} \int_a^b \left[ R_T(\lambda+i\epsilon)) - R_T(\lambda-i\epsilon) \right] \, d\lambda = \Omega_T((a,b)) + \frac{1}{2}\left[ \Omega_T(\{a\}) + \Omega_T(\{b\}) \right].</math>

Depending on the source, the '''resolution of the identity''' is defined, either as a projection-valued measure <math>\Omega_T</math>,<ref>{{cite book | last=Rudin | first=Walter | title=Functional Analysis | publisher=McGraw-Hill Science, Engineering & Mathematics | publication-place=Boston, Mass. | date=1991 | isbn=978-0-07-054236-5|pages=316–317}}</ref> or as a one-parameter family of projection-valued measures <math>\{\Sigma_\lambda\}</math> with <math>-\infty < \lambda < \infty</math>.<ref>{{cite book | last=Akhiezer | first=Naum Ilʹich | title=Theory of Linear Operators in Hilbert Space | publisher=Pitman | publication-place=Boston | date=1981 | isbn=0-273-08496-8|page=213}}</ref>

In the case of a discrete measure (in particular, when ''H'' is finite-dimensional), <math display="inline">I = \int 1\,d\Omega_T</math> can be written as
<math display="block">I = \sum_{i} \left | i \right \rangle  \left \langle i \right |</math>
in the Dirac notation, where each <math>|i\rangle</math> is a normalized eigenvector of ''T''. The set <math> \{ |i\rangle \}</math> 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
<math display="block">I = \int\!\!   di~ |i\rangle \langle i|</math>
and speak of a "continuous basis", or "continuum of basis states", <math> \{ |i\rangle \}</math> Mathematically, unless rigorous justifications are given, this expression is purely formal.

==References==
{{Reflist}}

{{Functional analysis}}
{{Spectral theory}}
{{Analysis in topological vector spaces}}

[[Category:Functional calculus]]