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The essential norm of an operator is not self-dual. (English) Zbl 0838.47010

Summary: It is established by an example that the distance of a bounded linear operator \(S\) from the class of compact operators on a Banach space is not always uniformly comparable with that of its adjoint \(S'\). This provides a negative solution to an old problem. It is also shown that the seminorms due to Schechter and Weis, that measure the deviation from strict singularity or strict cosingularity of an operator, are not uniformly comparable with the corresponding distance functions. Both results rely on a general construction related to certain approximation properties that are associated with closed ideals of operators.

MSC:

47A58 Linear operator approximation theory
46B28 Spaces of operators; tensor products; approximation properties
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