Part of the *Memoirs of the AMS *series, *Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory *is co-authored by Raúl E. Curto, Professor of Mathematics and Executive Associate Dean in the University of Iowa College of Liberal Arts and Sciences; In Sung Hwang of Sungkyunkwan University; and Woo Young Lee of Seoul National University.

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∞-functional calculus to an H∞+H∞-functional calculus for the compressions of the shift. Next, we consider the subnormality of Toeplitzoperators with matrix-valued bounded type symbols and, in particular, the matrix-valued version of Halmos’ Problem 5; we then establish a matrix-valued version of Abrahamse’s Theorem. We also solve a subnormal Toeplitz completion problem of 2×2 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.