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Matrix Completions, Moments, and Sums of Hermitian Squares$
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Mihály Bakonyi and Hugo J. Woerdeman

Print publication date: 2011

Print ISBN-13: 9780691128894

Published to University Press Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691128894.001.0001

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date: 14 December 2017

Contractive analogs

Contractive analogs

Chapter:
(p.257) Chapter Four Contractive analogs
Source:
Matrix Completions, Moments, and Sums of Hermitian Squares
Author(s):

Mihály Bakonyi

Hugo J. Woerdeman

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691128894.003.0004

This chapter deals with contractive completions of partial operator matrices. Since the norm of a submatrix is always less or equal to the norm of the matrix itself, every partial matrix which admits a contractive completion has to be partially contractive (or a partial contraction), that is, all its fully specified submatrices are contractions. The discussions cover contractive operator-matrix completions; linearly constrained completion problems; the operator-valued Nehari and Carathéodory problems; Nehari's problem in two variables; Nehari and Carathéodory problems for functions on compact groups; the Nevanlinna–Pick problem; the operator Corona problem; joint operator/Hilbert–Schmidt norm control extensions; an L1 extension problem for polynomials; superoptimal completions and approximations of analytic functions; and model matching. Exercises and notes are provided at the end of the chapter.

Keywords:   contractive completions, partial operator matrices, Nehari problem, Carathéodory problem, Nevanlinna–Pick problem, Corona problem, Hilbert–Schmidt norm control

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