Operator Lipschitz functions on Banach spaces

Jan Rozendaal; Fedor Sukochev; Anna Tomskova

Studia Mathematica (2016)

  • Volume: 232, Issue: 1, page 57-92
  • ISSN: 0039-3223

Abstract

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Let X, Y be Banach spaces and let (X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on (X,Y) and apply this theory to obtain commutator estimates of the form | | f ( B ) S - S f ( A ) | | ( X , Y ) ≤ c o n s t | | B S - S A | | ( X , Y ) for a large class of functions f, where A ∈ (X), B ∈ (Y) are scalar type operators and S ∈ (X,Y). In particular, we establish this estimate for f(t): = |t| and for diagonalizable operators on X = ℓ p and Y = ℓ q for p < q. We also study the estimate above in the setting of Banach ideals in (X,Y). The commutator estimates we derive hold for diagonalizable matrices with a constant independent of the size of the matrix.

How to cite

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Jan Rozendaal, Fedor Sukochev, and Anna Tomskova. "Operator Lipschitz functions on Banach spaces." Studia Mathematica 232.1 (2016): 57-92. <http://eudml.org/doc/285670>.

@article{JanRozendaal2016,
abstract = {Let X, Y be Banach spaces and let (X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on (X,Y) and apply this theory to obtain commutator estimates of the form $||f(B)S - Sf(A)||_\{(X,Y)\} ≤ const||BS - SA||_\{(X,Y)\}$ for a large class of functions f, where A ∈ (X), B ∈ (Y) are scalar type operators and S ∈ (X,Y). In particular, we establish this estimate for f(t): = |t| and for diagonalizable operators on $X = ℓ_\{p\}$ and $Y = ℓ_\{q\}$ for p < q. We also study the estimate above in the setting of Banach ideals in (X,Y). The commutator estimates we derive hold for diagonalizable matrices with a constant independent of the size of the matrix.},
author = {Jan Rozendaal, Fedor Sukochev, Anna Tomskova},
journal = {Studia Mathematica},
keywords = {operator Lipschitz function; double operator integral; diagonalizable operator; operator ideal},
language = {eng},
number = {1},
pages = {57-92},
title = {Operator Lipschitz functions on Banach spaces},
url = {http://eudml.org/doc/285670},
volume = {232},
year = {2016},
}

TY - JOUR
AU - Jan Rozendaal
AU - Fedor Sukochev
AU - Anna Tomskova
TI - Operator Lipschitz functions on Banach spaces
JO - Studia Mathematica
PY - 2016
VL - 232
IS - 1
SP - 57
EP - 92
AB - Let X, Y be Banach spaces and let (X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on (X,Y) and apply this theory to obtain commutator estimates of the form $||f(B)S - Sf(A)||_{(X,Y)} ≤ const||BS - SA||_{(X,Y)}$ for a large class of functions f, where A ∈ (X), B ∈ (Y) are scalar type operators and S ∈ (X,Y). In particular, we establish this estimate for f(t): = |t| and for diagonalizable operators on $X = ℓ_{p}$ and $Y = ℓ_{q}$ for p < q. We also study the estimate above in the setting of Banach ideals in (X,Y). The commutator estimates we derive hold for diagonalizable matrices with a constant independent of the size of the matrix.
LA - eng
KW - operator Lipschitz function; double operator integral; diagonalizable operator; operator ideal
UR - http://eudml.org/doc/285670
ER -

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