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 $$\Vert D{\Vert }_{{\ell }^p(I)\to {\ell }^p(I)}=1⟺\hat{I}˜(D^{\ast }D)=1,$$ where $\hat{I}˜$ 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.