Summing inclusion maps between symmetric sequence spaces

Abstract

In 1973/74 Bennett and (independently) Carl proved that for 1 ≤ u ≤ 2 1 \le u \le 2 the identity map id: ℓ u ↪ ℓ 2 \ell _u \hookrightarrow \ell _2 is absolutely ( u , 1 ) (u,1) -summing, i. e., for every unconditionally summable sequence ( x n ) (x_n) in ℓ u \ell _u the scalar sequence ( ‖ x n ‖ ℓ 2 ) (\|x_n \|_{\ell _2}) is contained in ℓ u \ell _u , which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a 2 2 -concave symmetric Banach sequence space E E the identity map id : E ↪ ℓ 2 \text {id}: E \hookrightarrow \ell _2 is absolutely ( E , 1 ) (E,1) -summing, i. e., for every unconditionally summable sequence ( x n ) (x_n) in E E the scalar sequence ( ‖ x n ‖ ℓ 2 ) (\|x_n \|_{\ell _2}) is contained in E 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 T on ℓ 2 \ell _2 with values in a 2 2 -concave symmetric Banach sequence space E E is a multiplier from ℓ 2 \ell _2 into E E . Furthermore, we prove an asymptotic formula for the k k -th approximation number of the identity map id : ℓ 2 n ↪ E n \text {id}: \ell _2^n \hookrightarrow E_n , where E n E_n denotes the linear span of the first n n standard unit vectors in E E , and apply it to Lorentz and Orlicz sequence spaces.