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.
References
- V. M. Adamjan, D. Z. Arov, and M. G. Kreĭn, Infinite Hankel matrices and generalized problems of Carathéodory-Fejér and F. Riesz, Funkcional. Anal. i Priložen. 2 (1968), no. 1, 1–19 (Russian). MR 0234274
- R. B. Alexeev and V. V. Peller, Badly approximable matrix functions and canonical factorizations, Indiana Univ. Math. J. 49 (2000), no. 4, 1247–1285. MR 1836530, DOI 10.1512/iumj.2000.49.1912
- Michael Cambern, Analytic range functions, J. Math. Anal. Appl. 12 (1965), 413–424. MR 188821, DOI 10.1016/0022-247X(65)90009-0
- Ronald G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, Vol. 49, Academic Press, New York-London, 1972. MR 0361893
- Henry Helson, Lectures on invariant subspaces, Academic Press, New York-London, 1964. MR 0171178
- Henry Helson and David Lowdenslager, Prediction theory and Fourier series in several variables. II, Acta Math. 106 (1961), 175–213. MR 176287, DOI 10.1007/BF02545786
- S. Ya. Havinson, On some extremal problems of the theory of analytic functions, Moskov. Gos. Univ. Učenye Zapiski Matematika 148(4) (1951), 133–143 (Russian). MR 0049322
- N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223, DOI 10.1007/978-3-642-70151-1
- V. V. Peller, Approximation by analytic operator-valued functions, Harmonic analysis and operator theory (Caracas, 1994) Contemp. Math., vol. 189, Amer. Math. Soc., Providence, RI, 1995, pp. 431–448. MR 1347029, DOI 10.1090/conm/189/02279
- Vladimir V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1949210, DOI 10.1007/978-0-387-21681-2
- V. V. Peller and S. R. Treil′, Superoptimal singular values and indices of infinite matrix functions, Indiana Univ. Math. J. 44 (1995), no. 1, 243–255. MR 1336440, DOI 10.1512/iumj.1995.44.1986
- V. V. Peller and S. R. Treil, Approximation by analytic matrix functions: the four block problem, J. Funct. Anal. 148 (1997), no. 1, 191–228. MR 1461500, DOI 10.1006/jfan.1996.3073
- —, Very badly approximable matrix functions, Selecta Math. New Ser. 11 (2005), 127–154.
- V. V. Peller and N. J. Young, Superoptimal analytic approximations of matrix functions, J. Funct. Anal. 120 (1994), no. 2, 300–343. MR 1266312, DOI 10.1006/jfan.1994.1034
- V. V. Peller and N. J. Young, Superoptimal singular values and indices of matrix functions, Integral Equations Operator Theory 20 (1994), no. 3, 350–363. MR 1299893, DOI 10.1007/BF01205287
- S. J. Poreda, A characterization of badly approximable functions, Trans. Amer. Math. Soc. 169 (1972), 249–256. MR 306510, DOI 10.1090/S0002-9947-1972-0306510-7
- Serguei Treil, On superoptimal approximation by analytic and meromorphic matrix-valued functions, J. Funct. Anal. 131 (1995), no. 2, 386–414. MR 1345037, DOI 10.1006/jfan.1995.1094
- V. I. Vasyunin, Formula for multiplicity of contractions with finite defect indices, Toeplitz operators and spectral function theory, Oper. Theory Adv. Appl., vol. 42, Birkhäuser, Basel, 1989, pp. 281–304. MR 1030054, DOI 10.1007/978-3-0348-5587-7_{6}
Bibliographic Information
- V. V. Peller
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
- MR Author ID: 194673
- Email: peller@math.msu.edu
- S. R. Treil
- Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
- MR Author ID: 232797
- Email: treil@math.brown.edu
- 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.
- © Copyright 2006 American Mathematical Society
- Journal: St. Petersburg Math. J. 17 (2006), 493-510
- MSC (2000): Primary 30D55, 47S35, 30E10, 46E40
- DOI: https://doi.org/10.1090/S1061-0022-06-00917-4
- MathSciNet review: 2167849