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Seminorms related to weak compactness and to Tauberian operators

Published online by Cambridge University Press:  24 October 2008

Kari Astala  and
Hans-Olav Tylli
Kari Astala
Affiliation:
Department of Mathematics, University of Helsinki, Hallituskatu 15, SF-00100 Helsinki, Finland
Hans-Olav Tylli
Affiliation:
Department of Mathematics, University of Helsinki, Hallituskatu 15, SF-00100 Helsinki, Finland
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Extract

Let E and F be Banach spaces. The semigroup Φ+(E,F) of semi-Fredholm operators consists of the bounded linear mappings E→F with closed image and finite-dimensional kernel. By a well known result of Yood we have that T∈Φ+(E,F) if and only if for any bounded set B⊂E the condition TB relatively compact implies that B is relatively compact. Lebow and Schechter[10] gave a quantitative version of the above qualitative characterization, namely the operator T belongs to Φ+(E,F) if and only if there is c ≥ 0 such that

for all bounded B⊂E. Here γ is the well known Hausdorff measure of non-compactness

with BE the closed unit ball of E.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 2 , March 1990 , pp. 367 - 375
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

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