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Operator Lipschitz functions on Banach spaces

Volume 232 / 2016

Jan Rozendaal, Fedor Sukochev, Anna Tomskova Studia Mathematica 232 (2016), 57-92 MSC: Primary 47A55, 47A56; Secondary 47B47. DOI: 10.4064/sm8499-3-2016 Published online: 14 April 2016


Let $X$, $Y$ be Banach spaces and let ${\mathcal {L}}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. We develop the theory of double operator integrals on ${\mathcal {L}}(X,Y)$ and apply this theory to obtain commutator estimates of the form $$ \| f(B)S-Sf(A)\| _{{\mathcal {L}}(X,Y)}\leq {\rm const}\,\| BS-SA\| _{{\mathcal {L}}(X,Y)} $$ for a large class of functions $f$, where $A\in {\mathcal {L}}(X)$, $B\in {\mathcal {L}}(Y)$ are scalar type operators and $S\in {\mathcal {L}}(X,Y)$. In particular, we establish this estimate for $f(t):=| t| $ and for diagonalizable operators on $X=\ell _{p}$ and $Y=\ell _{q}$ for $p \lt q$.

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


  • Jan RozendaalInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-656 Warszawa, Poland
  • Fedor SukochevSchool of Mathematics & Statistics
    University of NSW
    Kensington, NSW 2052, Australia
  • Anna TomskovaSchool of Mathematics & Statistics
    University of NSW
    Kensington, NSW 2052 Australia

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