Column extreme multipliers of the Free Hardy Space

Abstract

The full Fock space over $\mathbb C ^d$ can be identified with the free Hardy space, $H^2 (\mathbb B ^d _\mathbb N)$ - the unique non-commutative reproducing kernel Hilbert space corresponding to a non-commutative Szeg\"{o} kernel on the non-commutative, multi-variable open unit ball $\mathbb B ^d _\mathbb N := \bigsqcup _{n=1} ^\infty \left( \mathbb C^{n\times n} \otimes \mathbb C ^d \right) _1$. Elements of this space are free or non-commutative functions on $\mathbb B ^d _\mathbb N$. Under this identification, the full Fock space is the canonical non-commutative and several-variable analogue of the classical Hardy space of the disk, and many classical function theory results have faithful extensions to this setting. In particular to each contractive (free) multiplier $B$ of the free Hardy space, we associate a Hilbert space $\mathcal H(B)$ analogous to the deBranges-Rovnyak spaces in the unit disk, and consider the ways in which various properties of the free function $B$ are reflected in the Hilbert space $\mathcal H(B)$ and the operators which act on it. In the classical setting, the $\mathcal H(b)$ spaces of analytic functions on the disk display strikingly different behavior depending on whether or not the function $b$ is an extreme point in the unit ball of $H^\infty(\mathbb D)$. We show that such a dichotomy persists in the free case, where the split depends on whtether or not $B$ is what we call {\it column extreme}.