# 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)