Editing Borel functional calculus (section)

{{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]].