An elementary approach to an eigenvalue estimate for matrices (Q1977740)

A celebrated result of \textit{W. B. Johnson}, \textit{H. König}, \textit{B. Maurey} and \textit{J. R. Retherford} [Proc. int. Conf. Leipzig 1977, 100-105 (1978; Zbl 0408.47020)] is considered, which gives an eigenvalue estimate for any complex \(n\times n\) matrix \(T= (\tau_{ij})_{i,j}\): \[ \Biggl(\sum^n_{i= 1}|\lambda_i(T)|^p\Biggr)^{1/p}\leq \Biggl( \sum^n_{j=1} \Biggl( \sum^n_{i=1} |\tau_{ij}|^{p'}\Biggr)^{p/p'}\Biggr)^{1/p},\tag{1} \] where \(2\leq p<\infty\), \({1\over p}+{1\over p'}= 1\) and each eigenvalue \(\lambda_i(T)\) of \(T\) is counted according to its algebraic multiplicity. The original proof for \(p>2\) was given in the framework of absolutely \(p\)-summing operators in Banach spaces, using complex interpolation techniques and Brower's fixed point theorem. The aim of the present paper is to give a selfcontained ``elementary'' proof. An apparently weaker inequality is obtained by using the concept of entropy numbers and the entropy estimate of B. Maurey [cf. \textit{G. Pisier}, Remarques sur un résultat non publié de B. Maurey. Sémin. Anal. Fonct. 1980-1981, Exposé No.~5, 12 p. (1981; Zbl 0491.46017); \textit{M. Junge} and \textit{M. Defant}, Isr. J. Math. 84, No. 3, 417-433 (1993; Zbl 0781.41013)] for the convex hull of a set in a normed space. Then a tensor product trick is used in order to derive the stronger inequality (1).