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2009

Operator H\"older--Zygmund functions

Aleksandrov, A. B., Peller, V. V.

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It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz.

We show that the situation changes dramatically if we pass to H\"older classes.
Namely, we prove that if $f$ belongs to the H\"older class $\L_\a(\R)$ with $0<\a<1$, then $\|f(A)-f(B)\|\le\const\|A-B\|^\a$ for arbitrary self-adjoint operators $A$ and $B$.
We prove a similar result for functions $f$ in the Zygmund class $\L_1(\R)$: for arbitrary self-adjoint operators $A$ and $K$ we have $\|f(A-K)-2f(A)+f(A+K)\|\le\const\|K\|$.

Operator H\"older--Zygmund functions · Around

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1940
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V.V. Sten’kin
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