Certain numerical radius contraction operators

Furuta, Takayuki; Nakamoto, Ritsuo

doi:10.1090/S0002-9939-1971-0285924-2

Certain numerical radius contraction operators
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by Takayuki Furuta and Ritsuo Nakamoto PDF

Proc. Amer. Math. Soc. 29 (1971), 521-524 Request permission

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 $w(T) = \sup |(Tx,x)|$ 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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A strange dilation theorem

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