On a problem of J. P. Williams

Kissin, Edward; Shulman, Victor

doi:10.1090/S0002-9939-02-06608-X

On a problem of J. P. Williams
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by Edward Kissin and Victor S. Shulman PDF

Proc. Amer. Math. Soc. 130 (2002), 3605-3608 Request permission

Abstract:

Let $B(H)$ be the algebra of all bounded operators on a Hilbert space $H$. Let $g$ be a continuous function on the closed disk $D$ and let \[ \|g(A)X - Xg(A)\| \leq C\|AX - XA\|,\] for some $C > 0,$ for all $X \in B(H)$ and all $A \in B(H)$ with $\|A\|\leq 1$. Then $g$ is differentiable on $D$. The paper shows that the function $g$ may have a discontinuous derivative.

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