Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory

Abstract: In this paper, we study matrix functions of bounded type from the viewpoint of describing an interplay between function theory and operator theory. \ We first establish a criterion on the coprime-ness of two singular inner functions and obtain several properties of the Douglas-Shapiro-Shields factorizations of matrix functions of bounded type. \ We propose a new notion of tensored-scalar singularity, and then answer questions on Hankel operators with matrix-valued bounded type symbols. \ We also examine an interpolation problem related to a certain functional equation on matrix functions of bounded type; this can be seen as an extension of the classical Hermite-Fej\' er Interpolation Problem for matrix rational functions. \ We then extend the $H^\infty$-functional calculus to an $\overline{H^\infty}+H^\infty$-functional calculus for the compressions of the shift. \ Next, we consider the subnormality of Toeplitz operators with matrix-valued bounded type symbols and, in particular, the matrix-valued version of Halmos's Problem 5; we then establish a matrix-valued version of Abrahamse's Theorem. \ We also solve a subnormal Toeplitz completion problem of $2\times 2$ partial block Toeplitz matrices. \ Further, we establish a characterization of hyponormal Toeplitz pairs with matrix-valued bounded type symbols, and then derive rank formulae for the self-commutators of hyponormal Toeplitz pairs.