Summing inclusion maps between symmetric sequence spaces

Abstract:

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 $(\|x_n \|_{\ell _2})$ 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 $(\|x_n \|_{\ell _2})$ 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.