On the Theory of Matrix Valued Functions Belonging to the Smirnov Class

Details On the Theory of Matrix Valued Functions Belonging to the Smirnov Class On the Theory of Matrix Valued Functions Belonging to the Smirnov Class

2007-06-13

arxiv.org/abs/0706.1901v1

Topics in Interpolation Theory. Operator Theory: Advances and Applications, Vol. 95. Birkhauser 1997

Mathematics

Classical Analysis and ODEs

Scientific paper

A theory of matrix-valued functions from the matricial Smirnov class ${\goth N}_n^+({\Bbb D})$ is systematically developed. In particular, the maximum principle of V.I.Smirnov, inner-outer factorization, the Smirnov-Beurling characterization of outer functions and an analogue of Frostman's theorem are presented for matrix-valued functions from the Smirnov class ${\goth N}_n^+({\Bbb D})$. We also consider a family $F_{\lambda} =F-\lambda I$ of functions belonging to the matricial Smirnov class which is indexed by a complex parameter $\lambda$. We show that with the exception of a ''very small'' set of such $\lambda$ the corresponding inner factor in the inner-outer factorization of the function $F_{\lambda}$ is a Blaschke-Potapov product. The main goal of this paper is to provide users of analytic matrix-function theory with a standard source for references related to the matricial Smirnov class.

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