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Generalized Spectral Theory in Complex Banach Algebras

Published online by Cambridge University Press:  20 November 2018

G. N. Hile  and
W. E. Pfaffenberger
G. N. Hile
Affiliation:
University of Hawaii, Honolulu, Hawaii
W. E. Pfaffenberger
Affiliation:
University of Victoria, Victoria, British Columbia
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Let A be an element of a complex Banach algebra with identitI. The ordinary spectrum of A, sp(A), consists of those points z in the complex plane such that A — zI has no inverse in . If Q is any other element of , we define spQ(A), the spectrum of A relative to Q, or Q-spectrum of A, as those points z such that has no inverse in . Thus if Q = 0 the Q-spectrum of A is the same as the ordinary spectrum of A.

The generalized notion of spectrum, spQ(A), retains many of the properties of the ordinary spectrum, particularly when A and Q commute and the ordinary spectrum of Q does not meet the unit circle. Under these conditions the Q-spectrum of A is a nonempty compact subset of the plane, and if both sp(A) and sp(Q) are finite (or countable), so is spQ(A).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 6 , 01 December 1985 , pp. 1211 - 1236
Copyright
Copyright © Canadian Mathematical Society 1985

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