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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 Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691128894.001.0001

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Contractive analogs

Contractive analogs

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

Mihály Bakonyi

Hugo J. Woerdeman

Princeton University Press

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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