Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-15T01:36:57.290Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Apostol, C., On the roots of spectral operator-valued analytic junctions, Rev. Math. Pures Appl. 13 (1968), 587–589.Google Scholar
2. Berkson, E., A characterization of scalar operators on reflexive Banach spaces, Pacific J. Math. 13 (1963), 365–373.Google Scholar
3. Foguel, S. R., The relations between a spectral operator and its scalar part, Pacific J. Math. 8 (1958), 51–65.Google Scholar
4. Giles, J. R., Classes of semi-inner-product spaces, Trans. Amer. Math. Soc. 129 (1967) 436–446.Google Scholar
5. Koehler, D. O., A note on some operator theory in certain semi-inner product spaces (to appear).Google Scholar
6. Koehler, D. O. and Rosenthal, P., On isometries of normed linear spaces, Studia Math. 36 (1970), 215–218.Google Scholar
7. Lumer, G., Spectral operators, Hermitian operators, and bounded groups, Acta Sci. Math. 25 (1964), 75–85.Google Scholar
8. Stampfli, J. G., Adjoint abelian operators on Banach space, Can. J. Math. 21 (1969), 505–512.Google Scholar