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The perturbation of relatively open operators with reduced index

Published online by Cambridge University Press:  24 October 2008

L. E. Labuschagne
L. E. Labuschagne
Affiliation:
Department of Mathematics, University of Stellenbosch, 7600 Stellenbosch, South Africa
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Abstract

Let X and Y denote normed spaces and T:D(T) ⊂ X → Y a linear transformation. It is shown that even in the case where both X and Y are incomplete, the quantity remains constant under both small and compact perturbation, provided that T is relatively open, R(T) is closed, and the perturbation is made in the right ‘direction’. If in addition and N(T) is topologically complemented, the topological complementation of the kernel is also preserved under small perturbations made in the right ‘direction’ and arbitrary compact perturbation. Various counter-examples are exhibited proving these results to be best possible.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 2 , September 1992 , pp. 385 - 402
Copyright
Copyright © Cambridge Philosophical Society 1992

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