Orlicz norm estimates for eigenvalues of matrices

Abstract

Let φ be a supermultiplicative Orlicz function such that the function$t \mapsto \varphi \left( {\sqrt t } \right)$ is equivalent to a convex function. Then each complexn×n matrixT=(τ _ij ) _{i, j} satisfies the following eigenvalue estimate:$\left\| {\left( {\lambda _i \left( T \right)} \right)_{i = 1}^n } \right\|_{\ell _\varphi } \leqslant C\left\| ( \right\|\left( {\tau _{ij} } \right)_{i = 1}^n \left\| {_{_{\ell _{\varphi *} } } )_{j = 1}^n } \right\|\ell _{\bar \varphi }$. Here, ϕ_* stands for Young’s conjugate function of φ, ϕ,$\bar \varphi$ is the minimal submultiplicative function dominating φ andC>0 a constant depending only on φ. For the power function φ(t)=t ^p,p≥2 this is a celebrated result of Johnson, König, Maurey and Retherford from 1979. In this paper we prove the above result within a more general theory of related estimates.