Quasi-inner functions and local factors

We introduce the notion of {\it quasi-inner} function and show that the product $u=\rho_\infty\prod \rho_v$ of $m+1$ ratios of local {$L$-}factors {$\rho_v(z)=\gamma_v(z)/\gamma_v(1-z)$} over a finite set $F$ of places of the field of rational numbers {inclusive of} the archimedean place is {quasi-inner} on the left of the critical line $\Re(z)= \frac 12$ 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 non-archimedean ratios $\rho_v$ is quasi-inner and, in order to prove our main result we use Gauss multiplication theorem to factor the archimedean ratio $\rho_\infty$ into a product of $m$ quasi-inner functions whose product with each $\rho_v$ 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 $u=\rho_\infty$, and when $u(F)=\prod_{v\in F} \rho_v$ 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.