When does positivity of the operator-range kernel imply pointwise contractivity S(A)?

Ascertain whether, and under what conditions on A ∈ K and on the Schur multiplier S, the positivity of the kernel K(A,B) defined by K(A,B) = ∑_{n=0}^∞ A^n (I_p − S(A)S(B)^*) B^{* n} implies that the pointwise value S(A) is a contraction (i.e., ∥S(A)∥ ≤ 1). In particular, characterize the classes of A and S for which this implication holds beyond the scalar case A = a I_p.

Background

In Section 5.5 the authors develop an operator-range approach to co-isometric realizations and obtain a positive kernel K(A,B) that yields the structural identity K(A,B) − A K(A,B) B^* = I_p − S(A)S(B)^*. While in the scalar case A = a I_p positivity ensures pointwise contractivity of S(A), the authors note that for matrix arguments A this implication is not generally established.

They further adapt a known quaternionic example to produce a counterexample showing that, in general, positivity of the kernel does not force S(A) to be a contraction for some matrix A, highlighting the need to identify precise conditions under which the implication holds.