Operator Lipschitz functions on Banach spaces

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Operator Lipschitz functions on Banach spaces

Journal Article (2016)

Author(s)

Jan Rozendaal (TU Delft - Electrical Engineering, Mathematics and Computer Science)

Fedor Sukochev (The University of Newcastle, Australia)

Anna Tomskova (The University of Newcastle, Australia)

Research Group

Analysis

DOI related publication

https://doi.org/10.4064/sm8499-3-2016 Final published version

To reference this document use

https://resolver.tudelft.nl/uuid:55880866-3d12-480d-b2ea-00d28d98e7c1

More Info

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Publication Year

2016

Language

English

Research Group

Analysis

Issue number

1

Volume number

232

Pages (from-to)

57-92

Downloads counter

127

Abstract

Let X, Y be Banach spaces and let L(X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on L(X,Y) and apply this theory to obtain commutator estimates of the form

∥f(B)S−Sf(A)∥L(X,Y)≤const∥BS−SA∥L(X,Y)for a large class of functions f, where A∈L(X), B∈L(Y) are scalar type operators and S∈L(X,Y). In particular, we establish this estimate for f(t):=|t| and for diagonalizable operators on X=ℓp and Y=ℓq for p<q.

We also study the estimate above in the setting of Banach ideals in L(X,Y). The commutator estimates we derive hold for diagonalizable matrices with a constant independent of the size of the matrix.