Sub-Hardy Hilbert spaces in the non-commutative unit row-ball

Abstract

In the classical Hardy space theory of square-summable Taylor series in the complex unit disk there is a circle of ideas connecting Szegö's theorem, factorization of positive semi-definite Toeplitz operators, non-extreme points of the convex set of contractive analytic functions, de Branges-Rovnyak spaces and the Smirnov class of ratios of bounded analytic functions in the disk. We extend these ideas to the multi-variable and non-commutative setting of the full Fock space, identified as the free Hardy space of square-summable power series in several non-commuting variables. As an application, we prove a Fejér-Riesz style theorem for noncommutative rational functions.

Noncommutative rational Clark measures

“…In [39] we observed that any CE is necessarily an extreme point, and that if is non-CE, then one can define a unique Sarason outer function, ∈ [H ∞ ] 1 , so that (0) > 0 and := is column-extreme. In [42] we proved that is outer and that if = is NC rational and non-CE, then = is NC rational and the column := is inner. Of course it may be that either = 0 or = 0.…”

“…The Radon-Nikodym formula for the absolutely continuous (and pure) part of = in the theorem statement is established in [42,Theorem 6], and is a consequence of an NC rational Fejér-Riesz Theorem [42,Theorem 5] and the NC Fatou Theorem of [40]. It further follows from [20,Theorem 6.5] (see Theorem B) that Π = ⊕ =1 Π ( ) is the direct sum of finitely many irreducible Cuntz row isometries of dilation type.…”

“…* and := (0), see, for example,[42, Lemma 6]. Observe that H 0 ⊆ H is −invariant and that if H 0 ≠ H , then H 0 has co-dimension one.…”