Summing Inclusion Maps between Symmetric Sequence Spaces

In 1973/74 Bennett and (independently) Carl proved that for 1 ≤ u ≤ 2 the identity map id: `u ↪→ `2 is absolutely (u, 1)-summing, i. e., for every unconditionally summable sequence (xn) in `u the scalar sequence (‖xn‖`2 ) is contained in `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 : E ↪→ `2 is absolutely (E, 1)-summing, i. e., for every unconditionally summable sequence (xn) in E the scalar sequence (‖xn‖`2) 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 `2 with values in a 2-concave symmetric Banach sequence space E is a multiplier from `2 into E. Furthermore, we prove an asymptotic formula for the k-th approximation number of the identity map id : `2 ↪→ En, where En denotes the linear span of the first n standard unit vectors in E, and apply it to Lorentz and Orlicz sequence spaces.