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On a problem of J. P. Williams

Author(s): Edward Kissin; Victor S. Shulman
Journal: Proc. Amer. Math. Soc. 130 (2002), 3605-3608.
MSC (2000): Primary 47A56
Posted: May 8, 2002
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Abstract: Let be the algebra of all bounded operators on a Hilbert space . Let be a continuous function on the closed disk and let

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

for some

for all $X \in B(H)$ and all $A \in B(H)$ with $\Vert A\Vert\leq 1$ . Then

is differentiable on

. The paper shows that the function

may have a discontinuous derivative.

References:

1.: D. R. Jocic, Integral representation formula for generalized normal derivation, Proc. Amer. Math. Soc. 127 (1999), 2303-2314. MR 99j:47026
2.: B. E. Johnson and J. P. Williams, The range of a normal derivation, Pacific J. Math. 58 (1975), 105-122. MR 52:1390
3.: E. Kissin and V. S. Shulman, Classes of Operator-smooth Functions. I. Operator Lipschitz Functions, preprint, (2000).
4.: T. W. Palmer, Banach Algebras and the General Theory of -algebras, vol. I, CUP, 1994. MR 95c:46002
5.: R. Powers, A remark on the domain of an unbounded derivation of a -algebra, J. Funct. Anal. 18 (1975), 85-95. MR 52:1334
6.: B. Sz.-Nagy and C. Foias, Analyse Harmonique des Operateurs de l'espace de Hilbert, Academiai Kiado, Budapest, 1967. MR 37:778
7.: J. P. Williams, Derivation ranges: open problems, Topics in Modern Operator Theory, (Timisoara/Herculane, 1980), 319-328, Operator Theory: Adv. Appl., 2, Birkhäuser, Basel-Boston, MA, 1981. MR 83k:47027

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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: 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
Posted: May 8, 2002
Communicated by: David R. Larson
Copyright of article: Copyright 2002, American Mathematical Society


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