**Memoirs of the American Mathematical Society**

2019;
100 pp;
Softcover

MSC: Primary 30; 47;

**Print ISBN: 978-1-4704-3624-7
Product Code: MEMO/260/1253**

List Price: $81.00

AMS Member Price: $48.60

MAA Member Price: $72.90

**Electronic ISBN: 978-1-4704-5323-7
Product Code: MEMO/260/1253.E**

List Price: $81.00

AMS Member Price: $48.60

MAA Member Price: $72.90

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

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*Raúl E. Curto; In Sung Hwang; Woo Young Lee*

In this paper, the authors study matrix functions of bounded
type from the viewpoint of describing an interplay between function
theory and operator theory. They 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. They propose a new notion of
tensored-scalar singularity, and then answer questions on Hankel
operators with matrix-valued bounded type symbols. They 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.

The authors then extend the \(H^\infty\)-functional
calculus to an \(\overline{H^\infty}+H^\infty\)-functional calculus
for the compressions of the shift. Next, the authors consider the
subnormality of Toeplitz operators with matrix-valued bounded type
symbols and, in particular, the matrix-valued version of Halmos's
Problem 5 and then establish a matrix-valued version of Abrahamse's
Theorem. They also solve a subnormal Toeplitz completion problem of
\(2\times 2\) partial block Toeplitz matrices. Further, they 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.

#### Table of Contents

# Table of Contents

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

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 110
- Chapter 2. Preliminaries 716
- Chapter 3. Coprime inner functions 1322
- Chapter 4. Douglas-Shapiro-Shields factorizations 1928
- Chapter 5. Tensored-scalar singularity 3746
- Chapter 6. An interpolation problem and a functional calculus 4554
- Chapter 7. Abrahamse’s Theorem for matrix-valued symbols 5968
- Chapter 8. A subnormal Toeplitz completion 6776
- Chapter 9. Hyponormal Toeplitz pairs 7382
- Chapter 10. Concluding remarks 93102
- Bibliography 95104
- List of Symbols 99108
- Back Cover Back Cover1114