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We study Helson matrices (also known as multiplicative Hankel matrices), that is, infinite matrices of the form    M   (  α  )   =    {  α   (  n  m  )   }    n  ,  m  =  1   ∞    , where   α   is a sequence of complex numbers. Helson matrices are considered as linear operators onℓ  2    (  N  )    . By interpreting Helson matrices as Hankel matrices in countably many variables we use the theory of multivariate moment problems to show that    M  (  α  )    is non‐negative if and only if   α   is the moment sequence of a measure   μ   onR   ∞   , assuming that   α   does not grow too fast. We then characterize the non‐negative bounded Helson matrices    M  (  α  )    as those where the corresponding moment measures   μ   are Carleson measures for the Hardy space of countably many variables. Finally, we give a complete description of the Helson matrices of finite rank, in parallel with the classical Kronecker theorem on Hankel matrices.

On Helson matrices: moment problems, non‐negativity, boundedness, and finite rank