Donoghue-type extension for Schur multipliers over the unit spectral ball when p>1

Determine whether the following extension principle holds for matrix-size p>1: if a function S defined on a subset Ω of the unit spectral ball K = {Z ∈ C^{p×p} : ρ(Z) < 1} satisfies positivity of the Schur kernel K_S(A,B) = ∑_{n=0}^∞ A^n (I_p − S(A)S(B)^*) B^{* n} for all A,B ∈ Ω, then S is the restriction to Ω of a uniquely determined function on K of the form S(Z) = ∑_{n=0}^∞ Z^n S_n that is analytic in the sense of the paper and induces a contractive left ⋆-multiplication operator on H^2(K) (i.e., a Schur multiplier).

Background

Section 5 develops the notion of Schur multipliers S(Z) acting on the Hardy space H^2(K) over the unit spectral ball K via left ⋆-multiplication. The positivity of the kernel K_S(A,B) = ∑_{n=0}^∞ A^n (I_p − S(A)S(B)^*) B^{* n} on K characterizes such multipliers and implies that the associated operator is contractive.

In the classical scalar case (p=1), Donoghue’s analytic continuation theorem ensures that if the kernel is positive on any subset of the unit disk with an accumulation point, then the function extends uniquely to an analytic contractive function on the disk. The authors state that they do not know whether an analogous extension principle holds in the present noncommutative, matrix-variable setting when p>1.