Geometric and probabilistic estimates for entropy and approximation numbers of operators

Abstract

It is an open problem whether the entropy numbers en( T) of continuous linear operators T: X → Y are essentially self-dual, i.e., e n(T) ∼ e n(T ∗) . We give a positive result in the case that X and Y ∗ are of type 2, using volume estimates. This generalizes a result of Carl (On Gelfand, Kolmogorov, and entropy numbers of operators acting between special Banach spaces, University of Jena, Jena, East Germany, 1983, preprint). Moreover, we derive bounds for the approximation numbers a n ( T) of T by probabilistic averaging. The formulas are applied to determine the exact asymptotic order of the approximation numbers of the formal identity map between various sequence spaces as well as tensor product spaces. In the special case of l p n , the result was first proved by Gluskin ( Mat. Sb. 120 (1983) , 180–189. [Russian]) using a different method.