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In the classical Hardy space theory of square-summable Taylor series in thecomplex unit disk there is a circle of ideas connecting Szeg\"o's theorem,factorization of positive semi-definite Toeplitz operators, non-extreme pointsof the convex set of contractive analytic functions, de Branges--Rovnyak spacesand the Smirnov class of ratios of bounded analytic functions in the disk. Weextend these ideas to the multi-variable and non-commutative setting of thefull Fock space, identified as the \emph{free Hardy space} of square-summablepower series in several non-commuting variables. As an application, we prove aFej\'er-Riesz style theorem for non-commutative rational functions.

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