Operator Lipschitz functions on Banach spaces

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 $\Vert f(B)S-Sf(A){\Vert }_{\mathcal{L}(X,Y)}\le \text{const}\Vert BS-SA{\Vert }_{\mathcal{L}(X,Y)}$ 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 $p<q$ and $p=q=1$, and for $X=Y={\mathrm{c}}_0$. We also obtain results for $p\ge q$. 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.

arXiv commentFinal version published in Studia Mathematica, with some minor changes