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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Carathéodory-Toeplitz and Nehari problems
for matrix valued almost periodic functions


Authors: Leiba Rodman, Ilya M. Spitkovsky and Hugo J. Woerdeman
Journal: Trans. Amer. Math. Soc. 350 (1998), 2185-2227
MSC (1991): Primary 42A75, 26A99, 15A54, 47A68, 47A56, 47A57, 42A82, 47B35.
MathSciNet review: 1422908
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Abstract: In this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated. We present necessary and sufficient conditions for the existence of extensions in terms of Toeplitz and Hankel operators on Besicovitch spaces and Lebesgue spaces. Furthermore, when a solution exists a special extension (the band extension) is constructed which enjoys a maximum entropy property. A linear fractional parameterization of the set of all extensions is also provided. The techniques used in the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method.


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Additional Information

Leiba Rodman
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795
Email: lxrodm@math.wm.edu

Ilya M. Spitkovsky
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795
Email: ilya@math.wm.edu

Hugo J. Woerdeman
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795
Email: hugo@math.wm.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01937-0
PII: S 0002-9947(98)01937-0
Keywords: Almost periodic matrix functions, contractive extensions, positive extensions, canonical factorization, Besicovitch space, Hankel operators, Toeplitz operators, band method.
Received by editor(s): April 29, 1996
Received by editor(s) in revised form: September 18, 1996
Additional Notes: The research is partially supported by NSF Grants 9500924 (LR, HJW) and 9401848 (IMS). The research of IMS was also supported by a semester research grant from the College of William & Mary
Article copyright: © Copyright 1998 American Mathematical Society