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

By:

Curto, Raúl E

Contributor(s):

Material type: Text

TextSeries: Memoirs of the American Mathematical Society SeriesPublisher: Providence : American Mathematical Society, 2019Copyright date: ©2019Edition: 1st edDescription: 1 online resource (112 pages)Content type:

text

Media type:

computer

Carrier type:

online resource

ISBN:

9781470453237

Subject(s):

Genre/Form:

Electronic books.

Additional physical formats: Print version:: Matrix Functions of Bounded Type: an Interplay Between Function Theory and Operator TheoryLOC classification:

QA401 .C87 2019

Online resources:

Click to View

Contents:

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

Summary: 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.

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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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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