2014/11/07 by Sriram Balasubramanian, Balasubramanian, Sriram
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA
paper · pdf · doi:10.48550/arxiv.1411.1965
12 Pages
arxiv created 2014/11/07 · arxiv updated 2014/11/10
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.