Approximation Numbers of Identity Operators between Symmetric Sequence Spaces

Abstract

We prove asymptotic formulas for the behavior of approximation quantities of identity operators between symmetric sequence spaces. These formulas extend recent results of Defant, Mastylo, and Michels for identities lpn←Fn with an n-dimensional symmetric normed space Fn with p-concavity conditions on Fn and 1 ≤ p ≤ 2. We consider the general case of identities En←Fn with weak assumptions on the asymptotic behavior of the fundamental sequences of the n-dimensional symmetric spaces En and Fn. We give applications to Lorentz and Orlicz sequence spaces, again considerably generalizing results of Pietsch, Defant, Mastylo, and Michels.