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

Peller, V.; Treil, S.

doi:10.1090/S1061-0022-06-00917-4

Approximation by analytic operator functions. Factorizations and very badly approximable functions
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by V. V. Peller and S. R. Treil

St. Petersburg Math. J. 17 (2006), 493-510

DOI: https://doi.org/10.1090/S1061-0022-06-00917-4

Published electronically: March 21, 2006

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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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