Functional calculus

A homomorphism of a certain function algebra $ A $ (cf. Algebra of functions) into the algebra $ L ( X) $ of continuous linear operators on a topological vector space $ X $. 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, $ A $ is a topological (in particular, normed) function algebra on a certain subset $ K $ of the space $ \mathbf C ^ {n} $ containing the polynomials in the variables $ z ^ {1}, \dots, z ^ {n} $ (often as a dense subset), so that a functional calculus $ \phi : A \rightarrow L ( X) $ is a natural extension of the polynomial calculus $ p ( z ^ {1}, \dots, z ^ {n} ) \rightarrow p ( T _ {1}, \dots, T _ {n} ) $ in the commuting operators $ T _ {i} = \phi ( z ^ {i} ) $, $ 1 \leq i \leq n $; in this case one says that the collection $ T = ( T _ {1}, \dots, T _ {n} ) $ admits an $ A $-calculus and one writes $ \phi ( T) = f ( T) = f ( T _ {1}, \dots, T _ {n} ) $. An $ A $-calculus for $ T $ is a kind of spectral theorem, since the correspondence $ a \rightarrow \langle \phi ( a) x, x ^ \prime \rangle $, where $ x \in X $, $ x ^ \prime \in X ^ {*} $ and $ \langle , \rangle $ is the duality between $ X $ and $ X ^ {*} $, determines a weak operator-valued $ A $-distribution which commutes with $ T $.

The classical functional calculus of von Neumann–Murray–Dunford ( $ A = C ( K) $, $ X $ is a reflexive space) leads to the operator (projection) spectral measure

$$ \epsilon = \epsilon _ {T} : \ f ( T _ {1}, \dots, T _ {n} ) = \int\limits f d \epsilon . $$

The functional calculus of Riesz–Dunford ( $ n = 1 $, $ A = \mathop{\rm Hol} ( \sigma ( T)) $, that is, all functions holomorphic on the spectrum $ \sigma ( T) $ of the operator $ T $) leads to the formula

$$ f ( T) = \ { \frac{1}{2 \pi i } } \int\limits _ \gamma f ( \lambda ) R ( \lambda , T) d \lambda , $$

where $ R ( \lambda , T) = ( \lambda I - T) ^ {- 1} $ is the resolvent of $ T $ and $ \gamma $ is a contour enclosing $ \sigma ( T) $ inside and on which the function $ f $ is regular. Formulas of the latter type with several variables (operators) depend on the notation for a linear functional on $ \mathop{\rm Hol} ( \sigma ( T)) $ and on the way the joint spectrum $ \sigma ( T) $ of the collection $ T = ( T _ {1}, \dots, T _ {n} ) $ is defined (the size of the functional calculus also depends on the definition of $ \sigma ( T) $).

If $ T $ is a spectral operator, if $ S $ and $ N $ are its scalar and quasi-nilpotent parts, respectively, and if $ f \in \mathop{\rm Hol} ( \sigma ( T)) $, then the formula

$$ f ( T) = \ \sum _ {n \geq 0 } \frac{N ^ {n} }{n! } \int\limits _ {\sigma ( T) } f ^ { ( n) } d \epsilon , $$

where $ \epsilon $ is a resolution of the identity for $ T $, enables one to extend the Riesz–Dunford functional calculus for $ T $ to a wider class of functions. In particular, if $ N ^ {m + 1 } = 0 $, then $ T $ admits a functional calculus on the class $ C ^ {m} ( \sigma ( T)) $ of $ m $-times continuously-differentiable functions. If $ T $ is an operator of scalar type, then one can substitute bounded Borel functions on $ \sigma ( T) $ in this formula. In particular, the normal operators on a Hilbert space admit such a functional calculus. The converse is true: If an operator $ T $ 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 $ T $ 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 $ C \{ M _ {k} \} $-calculus was constructed for operators with a resolvent of sufficiently slow growth near the spectrum; this was based on the Carleman classes $ C ( \{ M _ {k} \} , \sigma ( T)) $ (cf. Quasi-analytic class) and used the formula

$$ f ( T) = - { \frac{1} \pi } {\int\limits \int\limits } _ {\mathbf C } \frac{\partial \widetilde{f} }{\partial \overline{z} } ( \lambda ) R ( \lambda , T) d \lambda \overline{ {d \lambda }} , $$

where $ \widetilde{f} $ is the so-called $ \overline \partial $-extension of the function $ f $ across the boundary of the spectrum $ \sigma ( T) $, that is, a $ C ^ {1} $-function with compact support in $ \mathbf C $ for which

$$ \left . f = \widetilde{f} \right | _ {\sigma ( T) } ,\ \ \left | \frac{\partial \widetilde{f} }{\partial \overline{z} } ( \lambda ) \right | \leq \ \textrm{ const } \cdot h _ {\{ M _ {k} \} } ( c \mathop{\rm dist} ( \lambda , K)). $$

Here

$$ \frac{\partial \widetilde{f} }{\partial \overline{z} } = \ { \frac{1}{2} } \left ( \frac{\partial \widetilde{f} }{\partial x } + i \frac{\partial \widetilde{f} }{\partial y } \right ) , $$

$$ h _ {\{ M _ {k} \} } ( r) = \inf _ { n } r ^ {n - 1 } \frac{M _ {n} }{n ! } , $$

and the operator $ T $ satisfies

$$ \| R ( \lambda , T) \| \leq \ \left ( \frac{h _ {\{ M _ {k} \} } ( \mathop{\rm dist} ( \lambda , K) ) }{| \mathop{\rm log} \mathop{\rm dist} ( \lambda , K) | } \right ) . $$

On the other hand, bounds on the operator polynomials $ p ( T) $ lead to more extensive calculi (than $ \mathop{\rm Hol} ( \sigma ( T)) $). For example, if $ X $ is a Hilbert space, then the von Neumann–Heinz inequality

$$ \| p ( T) \| \leq \ \max \{ {| p ( \xi ) | } : {| \xi | \leq \| T \| } \} $$

leads to the Szökefalvi-Nagy–Foias functional calculus ( $ A $ is the algebra of all holomorphic and bounded functions in the disc $ \{ {\xi \in \mathbf C } : {| \xi | < 1 } \} $, $ T $ 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 $ T $ is invariant under a linear similarity $ T \rightarrow V ^ {- 1} TV $ and can be used successfully to classify operators. In particular, there is an extensive theory of the so-called $ A $-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:

$$ \sigma ( f ( T)) = \ f ( \sigma ( T)),\ \ f \in A. $$

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 $ A $ contains a fine partition of unity (for example, if $ A = C ^ \infty $), then one can construct a local spectral analysis from an $ A $-functional calculus and, in particular, one can prove the existence of non-trivial invariant subspaces of the operator $ T $ (if $ \sigma ( T) $ contains more than one point); an example is an operator $ T $ (in a Banach space) with a spectrum that lies on a smooth curve $ \gamma $ and $ \int _ {0} ^ \infty \mathop{\rm log} ^ {+} \mathop{\rm log} ^ {+} \delta ( r) dr < \infty $, where $ \delta ( r) = \max \{ {\| R ( \lambda , T) \| } : { \mathop{\rm dist} ( \lambda , \gamma ) \geq r } \} $. 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)