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

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

Raul E Curto, In Sung Hwang and Woo Young Lee
Memoirs of the American Mathematical Society, Vol.260(1253), pp.1-112
07/01/2019
DOI: 10.1090/memo/1253
url
https://arxiv.org/pdf/1611.06462View
Open Access

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-Fejer Interpolation Problem for matrix rational functions. We then extend the H-infinity-functional calculus to an (H) over bar (infinity) + H-infinity-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' Problem 5; we then establish a matrix-valued version of Abrahamse's Theorem. We also solve a subnormal Toeplitz completion problem of 2 x 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.
 Mathematics Physical Sciences Science & Technology

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