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A Spectral Theory for Duality Systems of Operators on a Banach Space

Published online by Cambridge University Press:  20 November 2018

J. G. Stampfli
J. G. Stampfli*
Affiliation:
Indiana University, Bloomington, Indiana
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This note is an addendum to my earlier paper [8]. The class of adjoint abelian operators discussed there was small because the compatibility relation between the operator and the duality map was too restrictive. (In effect, the relation is appropriate for Hilbert space, but ill-suited for other Banach spaces where the unit ball is not round.) However, the techniques introduced in [8] permit us to readily obtain a spectral theory (of the Dunford type) for a wider class of operators on Banach spaces, as we shall show.

A duality system for the operator T is an ordered sextuple

(i) T is a bounded linear operator mapping the Banach space B into B,

(ii) ϕ is a duality map from B to B*. Thus, for x ∊ B, ϕ(x) = x* ∊ B*, where ‖x‖ = ‖x*‖ and x*(x) = ‖x‖2. The existence of ϕ follows easily from the Hahn-Banach Theorem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 5 , 01 October 1970 , pp. 994 - 996
Copyright
Copyright © Canadian Mathematical Society 1970

References

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