Upper bound of the norm of an infinite matrix with positive entries

Abstract

It is well known that there is no known formula for the norm of an
$n\times n$ matrix in terms of its entries if $n>4$. In this note, we sketch a proof for the norm of a $2\times 2$ matrix in terms of its entries by a different approach and we provide conditions for which give us the upper bound of the norm of an infinite matrix with positive entries. We propose a question concerning the boundedness of an infinite matrix with entries in the positive cone of a $C^*$-algebra.