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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Lipschitz functions of normal operators


Author: Fuad Kittaneh
Journal: Proc. Amer. Math. Soc. 94 (1985), 416-418
MSC: Primary 47B15; Secondary 47A60
MathSciNet review: 787884
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Abstract: It is shown that if $ N$ and $ M$ are normal operators on a separable, complex Hilbert space $ H$, and $ f$ is a Lipschitz function on $ \Omega = \sigma (N) \cup \sigma (M)$ (i.e., $ \left\vert {f(z) - f(w)} \right\vert \leqslant k\left\vert {z - w} \right\vert$ for some positive constant $ k$ and all $ z,w \in \Omega )$, then $ {\left\Vert {f(N)X - Xf(M)} \right\Vert _2} \leqslant k{\left\Vert {NX - XM} \right\Vert _2}$ for any operator $ X$ on $ H$. In particular, $ {\left\Vert {f(N) - f(M)} \right\Vert _2} \leqslant k{\left\Vert {N - M} \right\Vert _2}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0787884-4
PII: S 0002-9939(1985)0787884-4
Keywords: Lipschitz function, Hilbert-Schmidt operator, normal operator
Article copyright: © Copyright 1985 American Mathematical Society