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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., $\left | {f(z) - f(w)} \right | \leqslant k\left | {z - w} \right |$ for some positive constant $k$ and all $z,w \in \Omega )$, then ${\left \| {f(N)X - Xf(M)} \right \|_2} \leqslant k{\left \| {NX - XM} \right \|_2}$ for any operator $X$ on $H$. In particular, ${\left \| {f(N) - f(M)} \right \|_2} \leqslant k{\left \| {N - M} \right \|_2}$.

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Keywords: Lipschitz function, Hilbert-Schmidt operator, normal operator
Article copyright: © Copyright 1985 American Mathematical Society