Summing inclusion maps between symmetric sequence spaces

摘要

In 1973/74 Bennett and (independently) Carl proved that for 1less than or equal to2 the identity map id: l(u)hooked right arrowl(2) is absolutely (u,1)-summing, i.e., for every unconditionally summable sequence (x(n)) in l(u) the scalar sequence (parallel tox(n)parallel to(l2)) is contained in l(u), which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a 2-concave symmetric Banach sequence space E the identity map id : Ehooked right arrowl(2) is absolutely (E,1)-summing, i.e., for every unconditionally summable sequence (x(n)) in E the scalar sequence (parallel tox(n)parallel to(l2)) is contained in E. Various applications are given, e.g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator T on l(2) with values in a 2-concave symmetric Banach sequence space E is a multiplier from l(2) into E. Furthermore, we prove an asymptotic formula for the k-th approximation number of the identity map id : l(2)(n)hooked right arrowE(n), where E-n denotes the linear span of the first n standard unit vectors in E, and apply it to Lorentz and Orlicz sequence spaces. [References: 39]