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Extensions of a theorem of Fuglede and Putnam

Author: S. K. Berberian
Journal: Proc. Amer. Math. Soc. 71 (1978), 113-114
MSC: Primary 47B20; Secondary 47B10
MathSciNet review: 0487554
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Abstract: The operator equation $ AX = XB$ implies $ {A^ \ast }X = X{B^ \ast }$ when A and B are normal (theorem of Fuglede and Putnam). If X is of Hilbert-Schmidt class, the assumptions on A and B can be relaxed: it suffices that A and $ {B^ \ast }$ be hyponormal, or that B be invertible with $ \left\Vert A \right\Vert\left\Vert {{B^{ - 1}}} \right\Vert \leqslant 1$.

References [Enhancements On Off] (What's this?)

  • [1] S. K. Berberian, Note on a theorem of Fuglede and Putnam, Proc. Amer. Math. Soc. 10 (1959), 175-182. MR 0107826 (21:6548)
  • [2] -, Introduction to Hilbert space, Chelsea, New York, 1976.
  • [3] J. Dixmier, Les algèbres d'opérateurs dans l'espace hilbertien (Algèbres de von Neumann), 2nd ed., Gauthier-Villars, Paris, 1969.
  • [4] P. R. Halmos, A Hilbert space problem book, Springer-Verlag, New York, 1974. MR 675952 (84e:47001)
  • [5] F. Riesz and B. Sz.-Nagy, Leçons d'analyse fonctionelle, Akadémia Kiadó, Budapest, 1952.

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Article copyright: © Copyright 1978 American Mathematical Society