Factorization and approximation

problems for matrix functions

Author:
V. V. Peller

Journal:
J. Amer. Math. Soc. **11** (1998), 751-770

MSC (1991):
Primary 47B35, 30Dxx, 46Exx

DOI:
https://doi.org/10.1090/S0894-0347-98-00274-4

MathSciNet review:
1618768

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Abstract | References | Similar Articles | Additional Information

Abstract: We study maximizing vectors of Hankel operators with matrix-valued symbols. This study leads to a solution of the so-called recovery problem for unitary-valued functions and to a new approach to Wiener-Hopf factorizations for functions in a function space satisfying natural conditions. Finally, we improve earlier results of Peller and Young on hereditary properties of the operator of superoptimal approximation by analytic matrix functions.

**[AAK]**V. M. Adamjan, D. Z. Arov, and M. G. Kreĭn,*Infinite Hankel block matrices and related problems of extension*, Izv. Akad. Nauk Armjan. SSR Ser. Mat.**6**(1971), no. 2-3, 87–112 (Russian, with Armenian and English summaries). MR**0298454****[BG1]**M. S. Budjanu and I. C. Gohberg,*General theorems on the factorization of matrix-valued functions. I. The fundamental theorem*, Mat. Issled.**3**(1968), no. vyp. 2 (8), 87–103 (Russian). MR**0259609****[BG2]**M. S. Budjanu and I. C. Gohberg,*General theorems on the factorization of matrix-valued functions. I. The fundamental theorem*, Mat. Issled.**3**(1968), no. vyp. 2 (8), 87–103 (Russian). MR**0259609****[CG]**Kevin F. Clancey and Israel Gohberg,*Factorization of matrix functions and singular integral operators*, Operator Theory: Advances and Applications, vol. 3, Birkhäuser Verlag, Basel-Boston, Mass., 1981. MR**657762****[CJ]**Lennart Carleson and Sigvard Jacobs,*Best uniform approximation by analytic functions*, Ark. Mat.**10**(1972), 219–229. MR**322410**, https://doi.org/10.1007/BF02384810**[DG]**Harry Dym and Israel Gohberg,*Unitary interpolants, factorization indices and infinite Hankel block matrices*, J. Funct. Anal.**54**(1983), no. 3, 229–289. MR**724524**, https://doi.org/10.1016/0022-1236(83)90001-0**[G]**I. C. Gohberg,*The factorization problem in normed rings, functions of isometric and symmetric operators, and singular integral equations*, Uspehi Mat. Nauk**19**(1964), no. 1 (115), 71–124 (Russian). MR**0163184****[GK]**I. C. Gohberg and M. G. Kreĭn,*Systems of integral equations on the half-line with kernels depending on the difference of the arguments*, Uspehi Mat. Nauk (N.S.)**13**(1958), no. 2 (80), 3–72 (Russian). MR**0102720****[L]**P.B. LAX, Symmetrizable linear transformations,*Comm. Pure Appl. Math.***7**(1954), 633-647. MR**16:832d****[LS]**Georgii S. Litvinchuk and Ilia M. Spitkovskii,*Factorization of measurable matrix functions*, Operator Theory: Advances and Applications, vol. 25, Birkhäuser Verlag, Basel, 1987. Translated from the Russian by Bernd Luderer; With a foreword by Bernd Silbermann. MR**1015716****[M]**N.I. MUSKHELISHVILI,*Singular integral equation. Boundary problems of function theory and their application to mathematical physics*, 2nd Ed., Fizmatgiz, Moscow, 1962 (Russian); English transl. of 1st ed., Nordhoff, Groningen, 1953. MR**15:434e****[N]**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****[Pa]**M. Papadimitrakis,*On best uniform approximation by bounded analytic functions*, Bull. London Math. Soc.**28**(1996), no. 1, 15–18. MR**1356821**, https://doi.org/10.1112/blms/28.1.15**[Pe1]**V. V. Peller,*Description of Hankel operators of the class 𝔖_{𝔭} for 𝔭>0, investigation of the rate of rational approximation and other applications*, Mat. Sb. (N.S.)**122(164)**(1983), no. 4, 481–510 (Russian). MR**725454****[Pe2]**Vladimir V. Peller,*Hankel operators and multivariate stationary processes*, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 357–371. MR**1077396****[Pe3]**V. V. Peller,*Hankel operators and continuity properties of best approximation operators*, Algebra i Analiz**2**(1990), no. 1, 163–189 (Russian); English transl., Leningrad Math. J.**2**(1991), no. 1, 139–160. MR**1049909****[Pe4]**Vladimir V. Peller,*Boundedness properties of the operators of best approximation by analytic and meromorphic functions*, Ark. Mat.**30**(1992), no. 2, 331–343. MR**1289760**, https://doi.org/10.1007/BF02384879**[Pe5]**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**, https://doi.org/10.1090/conm/189/02279**[PK]**V. V. Peller and S. V. Khrushchëv,*Hankel operators, best approximations and stationary Gaussian processes*, Uspekhi Mat. Nauk**37**(1982), no. 1(223), 53–124, 176 (Russian). MR**643765****[PY1]**V. V. Peller and N. J. Young,*Superoptimal analytic approximations of matrix functions*, J. Funct. Anal.**120**(1994), no. 2, 300–343. MR**1266312**, https://doi.org/10.1006/jfan.1994.1034**[PY2]**V.V. PELLER AND N.J. YOUNG, Construction of superoptimal approximation,*Math. Control Signals Systems***8**(1995), 118-137.**[PY3]**V. V. Peller and N. J. Young,*Continuity properties of best analytic approximation*, J. Reine Angew. Math.**483**(1997), 1–22. MR**1431840**, https://doi.org/10.1515/crll.1997.483.1**[R]**Ju. A. Rozanov,*Statsionarnye sluchaĭ nye protsessy*, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR**0159363****[S]**I. B. Simonenko,*Certain general questions of the theory of the Riemann boundary value problem*, Izv. Akad. Nauk SSSR Ser. Mat.**32**(1968), 1138–1146 (Russian). MR**0235135****[To]**V. A. Tolokonnikov,*Generalized Douglas algebras*, Algebra i Analiz**3**(1991), no. 2, 231–252 (Russian); English transl., St. Petersburg Math. J.**3**(1992), no. 2, 455–476. MR**1137530****[Tr]**Serguei Treil,*On superoptimal approximation by analytic and meromorphic matrix-valued functions*, J. Funct. Anal.**131**(1995), no. 2, 386–414. MR**1345037**, https://doi.org/10.1006/jfan.1995.1094**[V]**N.P. VEKUA,*Systems of singular integral equations and some boundary problems*, GITTL, Moscow, 1950 (Russian); English transl., Noordhoff, Groningen, 1967. MR**13:247a**

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Additional Information

**V. V. Peller**

Affiliation:
Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

Email:
peller@math.ksu.edu

DOI:
https://doi.org/10.1090/S0894-0347-98-00274-4

Received by editor(s):
June 11, 1997

Additional Notes:
The author is partially supported by NSF grant DMS 9304011.

Article copyright:
© Copyright 1998
American Mathematical Society