Modulus of continuity of operator functions

Farforovskaya, Yu.; Nikolskaya, L.

doi:10.1090/S1061-0022-09-01058-9

Modulus of continuity of operator functions
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by Yu. B. Farforovskaya and L. Nikolskaya

St. Petersburg Math. J. 20 (2009), 493-506

DOI: https://doi.org/10.1090/S1061-0022-09-01058-9

Published electronically: April 8, 2009

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Abstract:

Let $A$ and $B$ be bounded selfadjoint operators on a separable Hilbert space, and let $f$ be a continuous function defined on an interval $[a,b]$ containing the spectra of $A$ and $B$. If $\omega _f$ denotes the modulus of continuity of $f$, then \[ \|f(A)-f(B)\| \leq 4\Big [\log \Big (\frac {b-a}{\|A-B\|}+1\Big )+1\Big ]^2 \cdot \omega _f(\|A-B\|).\] A similar result is true for unbounded selfadjoint operators, under some natural assumptions on the growth of $f$.

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