Operator Lipschitz functions on Banach spaces

TL;DR

This paper develops a theory of double operator integrals on Banach spaces to establish commutator estimates for operator functions, including the absolute value function, across various Banach space settings.

Contribution

It introduces a new framework for double operator integrals on Banach spaces and derives uniform commutator estimates for a broad class of functions and operators.

Findings

Established commutator estimates for $f(t)=|t|$ on $ ext{ell}_p$ and $ ext{ell}_q$ spaces.

Proved estimates are independent of matrix size for diagonalizable matrices.

Extended results to Banach ideals in operator 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 $\|f(B)S-Sf(A)\|_{\mathcal{L}(X,Y)}\leq \textrm{const} \|BS-SA\|_{\mathcal{L}(X,Y)}$ 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 $p<q$ and $p=q=1$, and for $X=Y=\mathrm{c}_{0}$. We also obtain results for $p\geq q$. 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.