Functional calculus
A homomorphism of a certain function algebra (cf. Algebra of functions) into the algebra
of continuous linear operators on a topological vector space
. 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,
is a topological (in particular, normed) function algebra on a certain subset
of the space
containing the polynomials in the variables
(often as a dense subset), so that a functional calculus
is a natural extension of the polynomial calculus
in the commuting operators
,
; in this case one says that the collection
admits an
-calculus and one writes
. An
-calculus for
is a kind of spectral theorem, since the correspondence
, where
,
and
is the duality between
and
, determines a weak operator-valued
-distribution which commutes with
.
The classical functional calculus of von Neumann–Murray–Dunford (,
is a reflexive space) leads to the operator (projection) spectral measure
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The functional calculus of Riesz–Dunford (,
, that is, all functions holomorphic on the spectrum
of the operator
) leads to the formula
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where is the resolvent of
and
is a contour enclosing
inside and on which the function
is regular. Formulas of the latter type with several variables (operators) depend on the notation for a linear functional on
and on the way the joint spectrum
of the collection
is defined (the size of the functional calculus also depends on the definition of
).
If is a spectral operator, if
and
are its scalar and quasi-nilpotent parts, respectively, and if
, then the formula
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where is a resolution of the identity for
, enables one to extend the Riesz–Dunford functional calculus for
to a wider class of functions. In particular, if
, then
admits a functional calculus on the class
of
-times continuously-differentiable functions. If
is an operator of scalar type, then one can substitute bounded Borel functions on
in this formula. In particular, the normal operators on a Hilbert space admit such a functional calculus. The converse is true: If an operator
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
is a spectral operator of scalar type (in a Hilbert space this is a linear operator that is similar to a normal operator).
In [5] the non-analytic -calculus was constructed for operators with a resolvent of sufficiently slow growth near the spectrum; this was based on the Carleman classes
(cf. Quasi-analytic class) and used the formula
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where is the so-called
-extension of the function
across the boundary of the spectrum
, that is, a
-function with compact support in
for which
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Here
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and the operator satisfies
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On the other hand, bounds on the operator polynomials lead to more extensive calculi (than
). For example, if
is a Hilbert space, then the von Neumann–Heinz inequality
![]() |
leads to the Szökefalvi-Nagy–Foias functional calculus ( is the algebra of all holomorphic and bounded functions in the disc
,
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 [8]).
Applications. The type of a functional calculus admitting an operator is invariant under a linear similarity
and can be used successfully to classify operators. In particular, there is an extensive theory of the so-called
-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:
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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 contains a fine partition of unity (for example, if
), then one can construct a local spectral analysis from an
-functional calculus and, in particular, one can prove the existence of non-trivial invariant subspaces of the operator
(if
contains more than one point); an example is an operator
(in a Banach space) with a spectrum that lies on a smooth curve
and
, where
. A corollary of the local analysis is Shilov's theorem on idempotents [2].
References
[1] | N. Dunford, J.T. Schwartz, "Linear operators" , 1–3 , Interscience (1958–1971) |
[2] | N. Bourbaki, "Elements of mathematics. Spectral theories" , Addison-Wesley (1977) (Translated from French) |
[3] | L. Waelbroeck, "Etude spectrale des algèbres complètes" Acad. Roy. Belgique Cl. Sci. , 31 : 7 (1960) |
[4] | J.L. Taylor, "The analytic-functional calculus for several commuting operators" Acta Math. , 125 : 1–2 (1970) pp. 1–38 |
[5] | 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) |
[6] | J. von Neumann, "Eine Spektraltheorie für allgemeine Operatoren eines unitären Räumes" Math. Nachr. , 4 (1950–1951) pp. 258–281 |
[7] | B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970) (Translated from French) |
[8] | 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 |
[9] | I. Colojoară, C. Foiaş, "Theory of generalized spectral operators" , Gordon & Breach (1968) |
[10] | Yu.I. Lyubich, V.I. Matsaev, "Operators with separable spectrum" Mat. Sb. , 56 : 2 (1962) pp. 433–468 (In Russian) |
[11] | J. Mikusiński, "Operational calculus" , Pergamon (1959) (Translated from Polish) |
[12] | V.P. Maslov, "Operational methods" , MIR (1976) (Translated from Russian) |
Comments
For a systematic treatment of analytic functional calculi in several variables cf. [a1].
References
[a1] | F.H. Vasilescu, "Analytic functional calculus and spectral decompositions" , Reidel & Ed. Academici (1982) |
Functional calculus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functional_calculus&oldid=14302