An Elementary Approach to an Eigenvalue Estimate for Matrices

Abstract

A celebrated result of Johnson, Maurey, König and Retherford from 1977 states that for $2 \leqslant p < \infty$ every complex $n \times n$ matrix $T = (\tau _{ij} )_{i,j}$ satisfies the following eigenvalue estimate:

$$\left( {\sum\limits_{i = 1}^n {\left| {\lambda _i \left( T \right)} \right|^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \leqslant \left( {\sum\limits_{j = 1}^n {\left( {\sum\limits_{i = 1}^n {\left| {\tau _{ij} } \right|^{p'} } } \right)} {p \mathord{\left/ {\vphantom {p {p'}}} \right. \kern-\nulldelimiterspace} {p'}}} \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \cdot $$

Based on the concept of entropy numbers and a simple product trick we give a selfcontained elementary proof.