Norm of infinite doubly stochastic matrices

In finite dimensions, every doubly stochastic matrix has the ℓ p -operator norm equal to 1 for all 1 ≤ p ≤ ∞ . However, in the infinite-dimensional setting, this property may fail since the norm can be strictly smaller than 1 when 1 < p < ∞ . 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 < ∞ , it is shown that ‖ D ‖ ℓ p ( I ) → ℓ p ( I ) = 1 ⇔ Θ ( D ∗ D ) = 1 , where Θ 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.