Norm of infinite doubly stochastic matrices

Abstract

In finite dimensions, every doubly stochastic matrix has the $\ell^p$-operator norm equal to $1$ for all $1 \le p \le \infty$. However, in the infinite-dimensional setting, this property may fail since the norm can be strictly smaller than $1$ when $1<p<\infty$. In this paper, a complete characterization of infinite doubly stochastic matrices for which the norm remains equal to $1$ is obtained. More precisely, for $1<p<\infty$, it is shown that $$ \|D\|_{\ell^p(I)\to\ell^p(I)}=1 \quad\iff\quad \Theta(D^*D)=1, $$ where $\Theta$ measures the maximal average mass of a finite square submatrix. Thus, the norm is equal to $1$ precisely when the matrix contains arbitrarily large finite regions in which it behaves almost like a finite doubly stochastic matrix. The proof uses a Cheeger-type argument, highlighting a natural connection with ideas from spectral graph theory.