Operator Lipschitz functions on Banach spaces

Jan Rozendaal¹ ; Fedor Sukochev² ; Anna Tomskova³

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland
² School of Mathematics & Statistics University of NSW Kensington, NSW 2052, Australia
³ School of Mathematics & Statistics University of NSW Kensington, NSW 2052 Australia

Studia Mathematica, Tome 232 (2016) no. 1, pp. 57-92

Résumé

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.

Détail

DOI : 10.4064/sm8499-3-2016

Keywords: banach spaces mathcal space bounded linear operators develop theory double operator integrals mathcal apply theory obtain commutator estimates form s sf mathcal leq const bs sa mathcal large class functions where mathcal mathcal scalar type operators mathcal particular establish estimate diagonalizable operators ell ell study estimate above setting banach ideals mathcal commutator estimates derive diagonalizable matrices constant independent size matrix