scholarly journals New Properties of the Multivariable $$H^\infty $$ Functional Calculus of Sectorial Operators

2021 ◽  
Vol 93 (4) ◽  
Author(s):  
Olivier Arrigoni ◽  
Christian Le Merdy
Keyword(s):  
Functional Calculus ◽  
Sectorial Operators

scholarly journals H ∞-functional calculus and models of Nagy–Foiaş type for sectorial operators

2010 ◽  
Vol 351 (3) ◽  
pp. 733-760 ◽  
Author(s):  
José E. Galé ◽  
Pedro J. Miana ◽  
Dmitry V. Yakubovich
Keyword(s):  
Functional Calculus ◽  
Sectorial Operators

scholarly journals The absolute functional calculus for sectorial operators

2005 ◽  
Author(s):  
◽  
Tamara Kucherenko
Keyword(s):  
Functional Calculus ◽  
Maximal Regularity ◽  
Real Interpolation ◽  
Future Research ◽  
Interpolation Spaces ◽  
Holomorphic Functional Calculus ◽  
Sectorial Operators ◽  
The Common ◽  
The Absolute ◽  
Key Theorem

We introduce the absolute functional calculus for sectorial operators. This notion is stronger than the common holomorphic functional calculus. We are able to improve a key theorem related to the maximal regularity problem and hence demonstrate the power and usefulness of our new concept. In trying to characterize spaces where sectorial operators have absolute calculus, we find that certain real interpolation spaces play a central role. We are then extending various known results in this setting. The idea of unifying theorems about sectorial operators on real interpolation spaces permeates our work and opens paths for future research on this subject.

scholarly journals FUNCTIONAL CALCULI FOR SECTORIAL OPERATORS AND RELATED FUNCTION THEORY

2021 ◽  
pp. 1-81
Author(s):  
Charles Batty ◽  
Alexander Gomilko ◽  
Yuri Tomilov
Keyword(s):  
Banach Spaces ◽  
Functional Calculus ◽  
Function Theory ◽  
Operator Norm ◽  
Spectral Mapping ◽  
Related Function ◽  
Norm Estimates ◽  
Sectorial Operators ◽  
Associated Function ◽  
Class Of Functions

Abstract We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them.

scholarly journals Counterexamples concerning powers of sectorial operators on a Hilbert space

1999 ◽  
Vol 60 (3) ◽  
pp. 459-468 ◽  
Author(s):  
Arnaud Simard
Keyword(s):  
Hilbert Space ◽  
Functional Calculus ◽  
Fundamental Result ◽  
Sectorial Operators ◽  
Explicit Constructions ◽  
Dimensional Hilbert Space ◽  
Imaginary Powers ◽  
Semigroup Of Contractions

We give explicit constructions of semigroups and operators with particular properties. First we build a bounded C0-semigroup which is invertible and which is not similar to a semigroup of contractions. Afterwards we exhibit operators which admit bounded imaginary powers of angle ω > 0 on a Hilbert space but which do not admit a bounded functional calculus on the sector of angle ω. (This gives the limit of McIntosh's fundamental result.) Finally we build, in the 2-dimensional Hilbert space, an operator which is not the negative generator of a semigroup of contractions, although its imaginary powers are bounded by eπ|s|/2.

scholarly journals Two Results About H∞ Functional Calculus on Analytic umd Banach Spaces

2003 ◽  
Vol 74 (3) ◽  
pp. 351-378 ◽  
Author(s):  
Christian Le Merdy
Keyword(s):  
Banach Space ◽  
Hardy Space ◽  
Banach Spaces ◽  
Functional Calculus ◽  
Sectorial Operators ◽  
Derivation Operator ◽  
Umd Banach Spaces ◽  
Vector Valued

AbstractLet X be a Banach space with the analytic UMD property, and let A and B be two commuting sectorial operators on X which admit bounded H∞ functional calculi with respect to angles θ1 and θ2 satisfying θ1 + θ2 > π. It was proved by Kalton and Weis that in this case, A + B is closed. The first result of this paper is that under the same conditions, A + B actually admits a bounded H∞ functional calculus. Our second result is that given a Banach space X and a number 1 ≦ p < ∞, the derivation operator on the vector valued Hardy space Hp (R; X) admits a bounded H∞ functional calculus if and only if X has the analytic UMD property. This is an ‘analytic’ version of the well-known characterization of UMD by the boundedness of the H∞ functional calculus of the derivation operator on vector valued Lp-spaces Lp (R; X) for 1 < p < ∞ (Dore-Venni, Hieber-Prüss, Prüss).

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