Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory (Memoirs of the American Mathematical Society) - Softcover

Raul E. Curto (author), In Sung Hwang (author) & Woo Young Lee (author)

Softcover

ISBN 10: 1470436248 ISBN 13: 9781470436247

Publisher: American Mathematical Society, 2019

Synopsis

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-Fejer 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.

About this title

About the Author

Raul E. Curto, University of Iowa, Iowa City.

In Sung Hwang, Sungkyunkwan University, Suwon, Korea.

Woo Young Lee, Seoul National University, Korea.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2019
• Language: English
• ISBN 10: 1470436248
• ISBN 13: 9781470436247
• Binding: Paperback
• Number of pages: 100