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Certain numerical radius contraction operators


Authors: Takayuki Furuta and Ritsuo Nakamoto
Journal: Proc. Amer. Math. Soc. 29 (1971), 521-524
MSC: Primary 47.40
DOI: https://doi.org/10.1090/S0002-9939-1971-0285924-2
MathSciNet review: 0285924
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Abstract: In this paper an operator T means a bounded linear operator on a complex Hilbert space H. The numerical radius norm $ w(T)$ of an operator T, is defined by $ w(T) = \sup \vert(Tx,x)\vert$ for every unit vector x in H. An operator T is said to be a numerical radius contraction if $ w(T) \leqq 1$. We shall give some theorems on certain numerical radius contraction operators and related results in consequence of these theorems.

Our central result is that an idempotent numerical radius contraction is a projection. Finally we prove that a periodic numerical radius contraction is the direct sum of zero and a unitary operator, that is to say, normal and partial isometric.


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  • [1] C. A. Berger, A strange dilation theorem, Notices Amer. Math. Soc. 12 (1965), 590. Abstract #625-152.
  • [2] T. Furuta, On the class of paranormal operators, Proc. Japan Acad. 43 (1967), 594-598. MR 36 #4354. MR 0221302 (36:4354)
  • [3] T. Furuta, M. Horie and R. Nakamoto, A remark on a class of operators, Proc. Japan Acad. 43 (1967), 607-609. MR 36 #4356. MR 0221304 (36:4356)
  • [4] T. Furuta and R. Nakamoto, Some theorems on certain contraction operators, Proc. Japan Acad., 45 (1969), 565-567. MR 40 #6294. MR 0253079 (40:6294)
  • [5] P. R. Halmos, Hilbert space problem book, Van Nostrand, Princeton, N. J., 1967. MR 34 #8178. MR 0208368 (34:8178)
  • [6] V. Istratescu, On some hyponormal operator, Pacific J. Math. 22 (1967), 413-417. MR 35 #4747. MR 0213893 (35:4747)
  • [7] V. Istratescu, T. Saito and T. Yoshino, On a class of operators, Tôhoku Math. J. (2) 18 (1966), 410-413. MR 35 #756. MR 0209860 (35:756)
  • [8] T. Kato, Remarks on the numerical radius (unpublished).
  • [9] C. Pearcy, An elementary proof of the power inequality for the numerical radius, Michigan Math. J. 13 (1966), 289-291. MR 34 #1853. MR 0201976 (34:1853)
  • [10] J. G. Stampfli, Normality and the numerical range of an operator, Bull. Amer. Math. Soc. 72 (1966), 1021-1022. MR 35 #3470. MR 0212599 (35:3470)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0285924-2
Keywords: Complex Hilbert space, bounded linear operator, numerical radius, numerical radius contraction, spectraloid, normaloid, paranormal operator, hyponormal operator, idempotent operator, periodic operator, partial isometric operator
Article copyright: © Copyright 1971 American Mathematical Society

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