Geodesic
Modulus of continuity of operator functions
Algebra i analiz, Tome 20 (2008) no. 3, pp. 224-242.

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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 $f$.
Keywords: Selfadjoint operator, operator function, modulas of continuity. 
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Yu. B. Farforovskaya; L. Nikolskaya. Modulus of continuity of operator functions. Algebra i analiz, Tome 20 (2008) no. 3, pp. 224-242. http://geodesic.mathdoc.fr/item/AA_2008_20_3_a8/

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