Quasi-inner functions and local factors

Abstract

We introduce the notion of quasi-inner function and show that the product u = ρ ∞ ρ v of m + 1 ratios of local L-factors ρ v (z) = γ v (z)/γ v (1 − z) over a finite set F of places of Q inclusive of the archimedean place is quasi-inner on the left of the critical line (z) = 1 2 in the following sense. The off diagonal part u 21 of the matrix of the multiplication by u in the orthogonal decomposition of the Hilbert space L 2 of square integrable functions on the critical line into the Hardy space H 2 and its orthogonal complement is a compact operator. When interpreted on the unit disk, the quasi-inner condition means that the associated Haenkel matrix is compact. We show that none of the individual nonarchimedean ratios ρ v is quasi-inner and, in order to prove our main result we use Gauss multiplication theorem to factor the archimedean ratio ρ ∞ into a product of m quasi-inner functions whose product with each ρ v retains the property to be quasi-inner. Finally we prove that Sonin's space is simply the kernel of the diagonal part u 22 for the quasi-inner function u = ρ ∞ , and when u(F ) = v∈F ρ v the kernels of the u(F ) 22 form an inductive system of infinite dimensional spaces which are the semi-local analogues of (classical) Sonin's spaces.