View source for Functional calculus

A homomorphism of a certain function algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420301.png" /> (cf. [[Algebra of functions|Algebra of functions]]) into the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420302.png" /> of continuous linear operators on a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420303.png" />. A functional calculus is one of the basic tools of general spectral analysis and the theory of Banach algebras and it enables one to use function-analytic methods in these disciplines. Usually, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420304.png" /> is a topological (in particular, normed) function algebra on a certain subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420305.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420306.png" /> containing the polynomials in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420307.png" /> (often as a dense subset), so that a functional calculus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420308.png" /> is a natural extension of the polynomial calculus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420309.png" /> in the commuting operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203011.png" />; in this case one says that the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203012.png" /> admits an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203014.png" />-calculus and one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203015.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203016.png" />-calculus for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203017.png" /> is a kind of spectral theorem, since the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203021.png" /> is the duality between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203023.png" />, determines a weak operator-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203024.png" />-distribution which commutes with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203025.png" />.

The classical functional calculus of von Neumann–Murray–Dunford (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203027.png" /> is a reflexive space) leads to the operator (projection) [[Spectral measure|spectral measure]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203028.png" /></td> </tr></table>

The functional calculus of Riesz–Dunford (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203030.png" />, that is, all functions holomorphic on the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203031.png" /> of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203032.png" />) leads to the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203033.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203034.png" /> is the [[Resolvent|resolvent]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203036.png" /> is a contour enclosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203037.png" /> inside and on which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203038.png" /> is regular. Formulas of the latter type with several variables (operators) depend on the notation for a linear functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203039.png" /> and on the way the joint spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203040.png" /> of the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203041.png" /> is defined (the size of the functional calculus also depends on the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203042.png" />).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203043.png" /> is a [[Spectral operator|spectral operator]], if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203045.png" /> are its scalar and quasi-nilpotent parts, respectively, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203046.png" />, then the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203047.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203048.png" /> is a [[Resolution of the identity|resolution of the identity]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203049.png" />, enables one to extend the Riesz–Dunford functional calculus for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203050.png" /> to a wider class of functions. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203052.png" /> admits a functional calculus on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203054.png" />-times continuously-differentiable functions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203055.png" /> is an operator of scalar type, then one can substitute bounded Borel functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203056.png" /> in this formula. In particular, the normal operators on a Hilbert space admit such a functional calculus. The converse is true: If an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203057.png" /> admits such a functional calculus (for operators in reflexive spaces it is sufficient to assume the existence of a functional calculus on the class of continuous functions), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203058.png" /> is a spectral operator of scalar type (in a Hilbert space this is a linear operator that is similar to a normal operator).

In [[#References|[5]]] the non-analytic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203059.png" />-calculus was constructed for operators with a resolvent of sufficiently slow growth near the spectrum; this was based on the Carleman classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203060.png" /> (cf. [[Quasi-analytic class|Quasi-analytic class]]) and used the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203061.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203062.png" /> is the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203064.png" />-extension of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203065.png" /> across the boundary of the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203066.png" />, that is, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203067.png" />-function with compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203068.png" /> for which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203069.png" /></td> </tr></table>

Here

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203070.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203071.png" /></td> </tr></table>

and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203072.png" /> satisfies

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203073.png" /></td> </tr></table>

On the other hand, bounds on the operator polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203074.png" /> lead to more extensive calculi (than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203075.png" />). For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203076.png" /> is a Hilbert space, then the von Neumann–Heinz inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203077.png" /></td> </tr></table>

leads to the Szökefalvi-Nagy–Foias functional calculus (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203078.png" /> is the algebra of all holomorphic and bounded functions in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203080.png" /> is a contraction without unitary parts), which has many applications in the theory of functional models for contraction operators. The analogue of the von Neumann–Heinz inequality for symmetric function spaces provides a functional calculus in terms of multipliers (of corresponding convolution spaces [[#References|[8]]]).

Applications. The type of a functional calculus admitting an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203081.png" /> is invariant under a linear similarity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203082.png" /> and can be used successfully to classify operators. In particular, there is an extensive theory of the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203084.png" />-scalar operators, which can be applied to many classes of operators and is not confined to classical spectral theory. For a successful use of a functional calculus it is expedient to have the so-called spectral-mapping theorems:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203085.png" /></td> </tr></table>

Such theorems have been proved for all the functional calculi listed above (after giving a suitable meaning to the right-hand side of the formula).

If the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203086.png" /> contains a fine partition of unity (for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203087.png" />), then one can construct a local spectral analysis from an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203088.png" />-functional calculus and, in particular, one can prove the existence of non-trivial invariant subspaces of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203089.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203090.png" /> contains more than one point); an example is an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203091.png" /> (in a Banach space) with a spectrum that lies on a smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203093.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203094.png" />. A corollary of the local analysis is Shilov's theorem on idempotents [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators" , '''1–3''' , Interscience  (1958–1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Spectral theories" , Addison-Wesley  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Waelbroeck,   "Etude spectrale des algèbres complètes"  ''Acad. Roy. Belgique Cl. Sci.'' , '''31''' :  7  (1960)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.L. Taylor,   "The analytic-functional calculus for several commuting operators"  ''Acta Math.'' , '''125''' :  1–2  (1970)  pp. 1–38</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.M. Dyn'kin,   "An operator calculus based on the Cauchy–Green formula"  ''Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst.'' , '''30'''  (1972)  pp. 33–39  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. von Neumann,   "Eine Spektraltheorie für allgemeine Operatoren eines unitären Räumes"  ''Math. Nachr.'' , '''4'''  (1950–1951)  pp. 258–281</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  B. Szökefalvi-Nagy,   Ch. Foiaş,   "Harmonic analysis of operators on Hilbert space" , North-Holland  (1970)  (Translated from French)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.V. Peller,   "Estimates of operator polynomials in symmetric spaces. Functional calculus for absolute contraction operators"  ''Math. Notes'' , '''25'''  (1979)  pp. 464–471  ''Mat. Zametki'' , '''25''' :  6  (1979)  pp. 899–912</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  I. Colojoară,   C. Foiaş,   "Theory of generalized spectral operators" , Gordon &amp; Breach  (1968)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  Yu.I. Lyubich,   V.I. Matsaev,   "Operators with separable spectrum"  ''Mat. Sb.'' , '''56''' :  2  (1962)  pp. 433–468  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  J. Mikusiński,   "Operational calculus" , Pergamon  (1959)  (Translated from Polish)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  V.P. Maslov,   "Operational methods" , MIR  (1976)  (Translated from Russian)</TD></TR></table>



====Comments====
For a systematic treatment of analytic functional calculi in several variables cf. [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.H. Vasilescu,   "Analytic functional calculus and spectral decompositions" , Reidel &amp; Ed. Academici  (1982)</TD></TR></table>