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

About this Title

Raúl E. Curto, Department of Mathematics, University of Iowa, Iowa City, IA 52242, U.S.A., In Sung Hwang, Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea and Woo Young Lee, Department of Mathematics and RIM, Seoul National University, Seoul 08826, Korea

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 260, Number 1253
ISBNs: 978-1-4704-3624-7 (print); 978-1-4704-5323-7 (online)
DOI: https://doi.org/10.1090/memo/1253
Published electronically: July 16, 2019
Keywords: Functions of bounded type, matrix functions of bounded type, coprime inner functions, Douglas-Shapiro-Shields factorizations, tensored-scalar singularity, compressions of the shifts, $H^\infty$-functional calculus, interpolation problems, Toeplitz operators, Hankel operators, subnormal, hyponormal, jointly hyponormal, Halmos’ Problem 5, Abrahamse’s Theorem, Toeplitz pairs
MSC: Primary 30J05, 30H10, 30H15, 47A13, 47A56, 47B20, 47B35; Secondary 30J10, 30J15, 30H35, 47A20, 47A57

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Preliminaries
  • 3. Coprime inner functions
  • 4. Douglas-Shapiro-Shields factorizations
  • 5. Tensored-scalar singularity
  • 6. An interpolation problem and a functional calculus
  • 7. Abrahamse’s Theorem for matrix-valued symbols
  • 8. A subnormal Toeplitz completion
  • 9. Hyponormal Toeplitz pairs
  • 10. Concluding remarks
  • List of Symbols

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-Fejér Interpolation Problem for matrix rational functions. We then extend the $H^\infty$-functional calculus to an $\bar {H^\infty }+H^\infty$-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’ 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.

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