Mathematics > Functional Analysis
[Submitted on 14 Jan 2015 (v1), last revised 21 Apr 2016 (this version, v2)]
Title:Operator Lipschitz functions on Banach spaces
View PDFAbstract: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.
Submission history
From: Jan Rozendaal Ph.D. [view email][v1] Wed, 14 Jan 2015 07:52:53 UTC (29 KB)
[v2] Thu, 21 Apr 2016 07:15:44 UTC (29 KB)
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