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Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory (Paperback)
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Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory (Paperback)
Series: Memoirs of the American Mathematical Society
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Total price: R2,069
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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-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.
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