Operator Lipschitz functions on Banach spaces
Volume 232 / 2016
Abstract
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.
