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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 a problem of J. P. Williams


Authors: Edward Kissin and Victor S. Shulman
Journal: Proc. Amer. Math. Soc. 130 (2002), 3605-3608
MSC (2000): Primary 47A56
Published electronically: May 8, 2002
MathSciNet review: 1920040
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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

\begin{displaymath}\Vert g(A)X - Xg(A)\Vert \leq C\Vert AX - XA\Vert,\end{displaymath}

for some $C > 0,$ for all $X \in B(H)$ and all $A \in B(H)$ with $\Vert A\Vert\leq 1$. Then $g$ is differentiable on $D$. The paper shows that the function $g$ may have a discontinuous derivative.


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Additional Information

Edward Kissin
Affiliation: School of Communications Technology and Mathematical Sciences, University of North London, Holloway, London N7 8DB, Great Britain
Email: e.kissin@unl.ac.uk

Victor S. Shulman
Affiliation: School of Communications Technology and Mathematical Sciences, University of North London, Holloway, London N7 8DB, Great Britain – and – Department of Mathematics, Vologda State Technical University, Vologda, Russia
Email: shulman_v@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06608-X
PII: S 0002-9939(02)06608-X
Received by editor(s): March 19, 2001
Received by editor(s) in revised form: July 6, 2001
Published electronically: May 8, 2002
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society