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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Certain numerical radius contraction operators

Authors: Takayuki Furuta and Ritsuo Nakamoto
Journal: Proc. Amer. Math. Soc. 29 (1971), 521-524
MSC: Primary 47.40
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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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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