Eigenvalue estimates for operators on symmetric Banach sequence spaces

Defant, Andreas; Mastyło, Mieczysław; Michels, Carsten

doi:10.1090/S0002-9939-03-07106-5

Eigenvalue estimates for operators on symmetric Banach sequence spaces
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by Andreas Defant, Mieczysław Mastyło and Carsten Michels PDF

Proc. Amer. Math. Soc. 132 (2004), 513-521 Request permission

Abstract:

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.

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