Eigenvalue estimates for operators on symmetric Banach sequence spaces

Using abstract interpolation theory, we study eigenvalue distribution problems for operators on complex symmetric Banach sequence spaces. More precisely, extending two well-known results due to König on the asymptotic eigenvalue distribution of operators on $\ell_p$-spaces, we prove an eigenvalue estimate for Riesz operators on $\ell_p$-spaces with $1 < p < 2$, which take values in a $p$-concave symmetric Banach sequence space $E \hookrightarrow \ell_p$, as well as a dual version, and show that each operator $T$ on a $2$-convex symmetric Banach sequence space $F$, which takes values in a $2$-concave symmetric Banach sequence space $E$, is a Riesz operator with a sequence of eigenvalues that forms a multiplier from $F$ into $E$. Examples are presented which among others show that the concavity and convexity assumptions are essential.