Orlicz norm estimates for eigenvalues of matrices

Let φ be a supermultiplicative Orlicz function such that the function\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$t \mapsto \varphi \left( {\sqrt t } \right)$$ \end{document} is equivalent to a convex function. Then each complexn×n matrixT=(τij)i, j satisfies the following eigenvalue estimate:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\| {\left( {\lambda _i \left( T \right)} \right)_{i = 1}^n } \right\|_{\ell _\varphi } \leqslant C\left\| ( \right\|\left( {\tau _{ij} } \right)_{i = 1}^n \left\| {_{_{\ell _{\varphi *} } } )_{j = 1}^n } \right\|\ell _{\bar \varphi } $$ \end{document}. Here, ϕ* stands for Young’s conjugate function of φ, ϕ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar \varphi $$ \end{document} is the minimal submultiplicative function dominating φ andC>0 a constant depending only on φ. For the power function φ(t)=tp,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.