Norm estimates for matrix operators between Banach spaces

Abstract We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille–Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.

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Continuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or Hörmander classes

Analysis and Applications ◽

10.1142/s0219530516500159 ◽

2017 ◽

Vol 15 (03) ◽

pp. 353-389 ◽

Cited By ~ 9

Author(s):

Joachim Toft

Keyword(s):

Banach Spaces ◽

Differential Operators ◽

Modulation Spaces ◽

Von Neumann ◽

Norm Estimates ◽

Pseudo Differential Operators

We deduce continuity and Schatten–von Neumann properties for operators with matrices satisfying mixed quasi-norm estimates with Lebesgue and Schatten parameters in [Formula: see text]. We use these results to deduce continuity and Schatten–von Neumann properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces, or in appropriate Hörmander classes.

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FUNCTIONAL CALCULI FOR SECTORIAL OPERATORS AND RELATED FUNCTION THEORY

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000414 ◽

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.

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Perturbation results in the Fredholm theory and m-essential spectra of some matrix operators

Filomat ◽

10.2298/fil2107141a ◽

2021 ◽

Vol 35 (7) ◽

pp. 2141-2149

Author(s):

Boulbeba Abdelmoumen ◽

Sadok Chakroun ◽

Mnif Maher

Keyword(s):

Banach Spaces ◽

Essential Spectrum ◽

Fredholm Theory ◽

Essential Spectra ◽

Matrix Operators

In this paper, we will use some new properties of non-compactness measure, in order to establish a description of the M-essential spectrum for some matrix operators on Banach spaces.

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New Conditions for the Exponential Stability of Pseudolinear Difference Equations in Banach Spaces

Abstract and Applied Analysis ◽

10.1155/2016/5098086 ◽

2016 ◽

Vol 2016 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Rigoberto Medina

Keyword(s):

Exponential Stability ◽

Banach Spaces ◽

Nonlinear Perturbations ◽

Small Perturbations ◽

Norm Estimates ◽

Infinite Dimensional ◽

Varying Coefficients ◽

Freezing Method ◽

Difference Systems ◽

Combined Use

We study the local exponential stability of evolution difference systems with slowly varying coefficients and nonlinear perturbations. We establish the robustness of the exponential stability in infinite-dimensional Banach spaces, in the sense that the exponential stability for a given pseudolinear equation persists under sufficiently small perturbations. The main methodology is based on a combined use of new norm estimates for operator-valued functions with the “freezing” method.

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Kazhdan projections, random walks and ergodic theorems

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2017-0002 ◽

2019 ◽

Vol 2019 (754) ◽

pp. 49-86 ◽

Cited By ~ 2

Author(s):

Cornelia Druţu ◽

Piotr W. Nowak

Keyword(s):

Random Walks ◽

Banach Spaces ◽

Ergodic Theorems ◽

Spectral Gaps ◽

Norm Estimates ◽

Markov Operators ◽

Open Questions ◽

Convergence Results ◽

Uniformly Convex Banach Spaces ◽

Property T

Abstract In this paper we investigate generalizations of Kazhdan’s property (T) to the setting of uniformly convex Banach spaces. We explain the interplay between the existence of spectral gaps and that of Kazhdan projections. Our methods employ Markov operators associated to a random walk on the group, for which we provide new norm estimates and convergence results. This construction exhibits useful properties and flexibility, and allows to view Kazhdan projections in Banach spaces as natural objects associated to random walks on groups. We give a number of applications of these results. In particular, we address several open questions. We give a direct comparison of properties (TE) and FE with Lafforgue’s reinforced Banach property (T); we obtain shrinking target theorems for orbits of Kazhdan groups; finally, answering a question of Willett and Yu we construct non-compact ghost projections for warped cones. In this last case we conjecture that such warped cones provide counterexamples to the coarse Baum–Connes conjecture.

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