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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

Approximation by analytic operator functions. Factorizations and very badly approximable functions


Authors: V. V. Peller and S. R. Treil
Original publication: Algebra i Analiz, tom 17 (2005), nomer 3.
Journal: St. Petersburg Math. J. 17 (2006), 493-510
MSC (2000): Primary 30D55, 47S35, 30E10, 46E40
Published electronically: March 21, 2006
MathSciNet review: 2167849
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Abstract | References | Similar Articles | Additional Information

Abstract: This is a continuation of our earlier paper published in Selecta Math. 11 (2005), 127-154. We consider here operator-valued functions (or infinite matrix functions) on the unit circle $ \mathbb{T}$ and study the problem of approximation by bounded analytic operator functions. We discuss thematic and canonical factorizations of operator functions and study badly approximable and very badly approximable operator functions.

We obtain algebraic and geometric characterizations of badly approximable and very badly approximable operator functions. Note that there is an important difference between the case of finite matrix functions and the case of operator functions. Our criteria for a function to be very badly approximable in the case of finite matrix functions also guarantee that the zero function is the only superoptimal approximant. However, in the case of operator functions this is not true.


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Additional Information

V. V. Peller
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: peller@math.msu.edu

S. R. Treil
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
Email: treil@math.brown.edu

DOI: http://dx.doi.org/10.1090/S1061-0022-06-00917-4
PII: S 1061-0022(06)00917-4
Keywords: Superoptimal approximation, badly approximable operator functions, very badly approximable operator functions, Hankel operators, Toeplitz operators
Received by editor(s): November 30, 2004
Published electronically: March 21, 2006
Additional Notes: The first author was partially supported by NSF grant DMS 0200712. The second author was partially supported by NSF grant DMS 0200584.
Article copyright: © Copyright 2006 American Mathematical Society