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Toeplitz Corona and the Douglas Property for Free Functions

The well known Douglas Lemma says that for operators $A,B$ on Hilbert space that $AA^*-BB^*\succeq 0$ implies $B=AC$ for some contraction operator $C$. The result carries over directly to classical operator-valued Toeplitz operators by simply replacing operator by Toeplitz operator. Free functions generalize the notion of free polynomials and formal power series and trace back to the work of J. Taylor in the 1970s. They are of current interest, in part because of their connections with free probability and engineering systems theory. For free functions $a$ and $b$ on a free domain $\cK$ defined free polynomial inequalities, a sufficient condition on the difference $aa^*-bb^*$ to imply the existence a free function $c$ taking contractive values on $\cK$ such that $a=bc$ is established. The connection to recent work of Agler and McCarthy and their free Toeplitz Corona Theorem is exposited.