Duality of the weak essential norm
Copyright policy )Duality of the weak essential norm
It is established by an example that the natural quotient norms S ↦ d i s t ( S , W ( E , F ) ) S \mapsto \mathrm {dist}(S,W(E,F)) and S ↦ d i s t ( S ∗ , W ( F ∗ , E ∗ ) ) S \mapsto \mathrm {dist}(S^{*},W(F^{*},E^{*})) are not comparable in general. Hence there is no uniform quantitative version of Gantmacher’s duality theorem for weakly compact operators in terms of the preceding weak essential norm. Above W ( E , F ) W(E,F) stands for the class of weakly compact operators E → F E\to F , where E E and F F are Banach spaces. The counterexample is based on a renorming construction related to weakly compact approximation properties that is applied to the Johnson-Lindenstrauss space J L JL .
- University of Helsinki Finland
Geometry and structure of normed linear spaces, Linear operators defined by compactness properties, Norms (inequalities, more than one norm, etc.) of linear operators, weakly compact operators, Spaces of operators; tensor products; approximation properties, weak essential norm
Geometry and structure of normed linear spaces, Linear operators defined by compactness properties, Norms (inequalities, more than one norm, etc.) of linear operators, weakly compact operators, Spaces of operators; tensor products; approximation properties, weak essential norm
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