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On the joint spectra of doubly commuting n-tuples of semi-normal operators

Published online by Cambridge University Press:  18 May 2009

Muneo Chō  and
A. T. Dash
Muneo Chō
Affiliation:
Joetsu University of Education, Department of Mathematics, Joeisu 943, Japan
A. T. Dash
Affiliation:
Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada
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Let H be a complex Hilbert space. For any operator (bounded linear transformation) T on H, we denote the spectrum of T by σ(T). Let T = (T1, …, Tn) be an n-tuple of commuting operators on H. Let Sp(T) be the Taylor joint spectrum of T. We refer the reader to [8] for the definition of Sp(T). A point v = (v1, …, vn) of ℂn is in the joint approximate point spectrum σπ(T) of T if there exists a sequence {xk} of unit vectors in H such that

.

A point v = (v1, …, vn) of ℂn is in the joint approximate compression spectrum σs(T) of T if there exists a sequence {xk} of unit vectors in H such that

A point v=(v1, …, vn) of ℂn is in the joint point spectrum σp(T) of T if there exists a non-zero vector x in H such that (Ti-vi)x = 0 for all i, 1 ≤ j ≤ n.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 26 , Issue 1 , January 1985 , pp. 47 - 50
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

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