Summing inclusion maps between symmetric sequence spaces

In 1973/74 Bennett and (independently) Carl proved that for $1 \le u \le 2$ the identity map id: $\ell_u \hookrightarrow \ell_2$ is absolutely $(u,1)$-summing, i.e., for every unconditionally summable sequence $(x_n)$in $\ell_u$ the scalar sequence $(\Vert x_n \Vert _{\ell_2})$ is contained in $\ell_u$, 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 $\text{id}: E \hookrightarrow \ell_2$ is absolutely $(E,1)$-summing, i.e., for every unconditionally summable sequence $(x_n)$ in $E$ the scalar sequence $(\Vert x_n \Vert _{\ell_2})$ 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 $\ell_2$ with values in a $2$-concave symmetric Banach sequence space $E$ is a multiplier from $\ell_2$ into $E$. Furthermore, we prove an asymptotic formula for the $k$-th approximation number of the identity map $\text{id}: \ell_2^n \hookrightarrow E_n$, 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.