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On a Family of Generalized Numerical Ranges

Published online by Cambridge University Press:  20 November 2018

C.-S. Lin
C.-S. Lin*
Affiliation:
University of New Brunswick, Fredericton, New Brunswick
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Throughout this note, an operator will always mean a bounded linear operator acting on a Hilbert space X into itself, unless otherwise stated. The class Cρ (0 < ρ < ∞ ) of operators, considered by Sz.-Nagy and Foiaş [5], is defined as follows: An operator T is in Cρ if Tnx = pPUnx for all x ∊ X, n = 1, 2, . . . , where U is a unitary operator on some Hilbert space Y containing X as a subspace, and P is the orthogonal projection of Y onto X. In [2] Holbrook defined the operator radii wρ(·) (0 < ρ ≦ ∞ ) as the generalized Minkowski distance functionals on the Banach algebra of bounded linear operators on X, i.e.,

and w∞(T) = r(T), the spectral radius of T [2, Theorem 5.1].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 3 , June 1974 , pp. 678 - 685
Copyright
Copyright © Canadian Mathematical Society 1974

References

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