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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: 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 $X$ 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.


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  • [AAK] V.M. ADAMYAN, D.Z. AROV AND M.G. KREIN, Infinite Hankel block matrices and some related continuation problems, Izv. Akad. Nauk Armyan. SSR Ser. Mat 6 (1971), 87-112. MR 45:7506
  • [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. 2(8), 87-103 (Russian); English transl., Amer. Math. Soc. Transl. (2) 102 (1973), 1-14. MR 41:4246a
  • [BG2] M.S. BUDJANU AND I.C. GOHBERG, General theorems on the factorization of matrix valued functions, II. Some tests and their consequences, Mat. Issled. 3 (1968), no. 3(8), 3-18 (Russian); English transl., Amer. Math. Soc. Transl. (2) 102 (1973), 15-26. MR 41:4246b
  • [CG] K. CLANCEY AND I. GOHBERG, Factorization of matrix functions and singular integral operators, Oper. Theory: Advances and Appl., 3, Birkhäuser Verlag, Basel-Boston-Stuttgart, 1981. MR 84a:47016
  • [CJ] L. CARLESON AND S. JACOBS, Best uniform approximation by analytic functions, Ark. Mat. 10 (1972), 219-229. MR 48:772
  • [DG] H. DYM AND I. C. GOHBERG, Unitary interpolants, factorization indices and infinite Hankel block matrices, J. Funct. Anal. 54 (1983), 229-289. MR 85m:47021a
  • [G] I.C. GOHBERG, The factorization problem in normed rings, functions of isometric and symmetric operators and singular integral equations, Uspekhi Mat. Nauk 19:1 (1964), 71-124 (Russian); English transl., Russian Math. Surveys 19:1 (1964), 63-114. MR 29:487
  • [GK] I.C. GOHBERG AND M.G. KREIN, Systems of integral equations on a half line with kernels depending on the difference of arguments, Uspekhi Mat. Nauk 13:2 (1958), 3-72 (Russian); English transl., Amer. Math. Soc. Transl. 14 (1960), 217-287. MR 21:1506
  • [L] P.B. LAX, Symmetrizable linear transformations, Comm. Pure Appl. Math. 7 (1954), 633-647. MR 16:832d
  • [LS] G.S. LITVINCHUK AND I.M. SPITKOVSKI, Factorization of measurable matrix functions, Oper. Theory: Advances and Appl., 25, Birkhäuser Verlag, Basel-Boston, MA, 1987. MR 90g:47030
  • [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'SKII, Treatise on the shift operator. Spectral function theory, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1986. MR 87i:47042
  • [Pa] M. PAPADIMITRAKIS, On best uniform approximation by bounded analytic functions, Bull. London Math. Soc. 28 (1996), 15-18. MR 96j:30054
  • [Pe1] V.V. PELLER, A description of Hankel operators of class ${\frak S}_p$ for $p>0$, investigation of the rate of rational approximation and other applications, Mat. Sb. 122 (1983), 481-510; English transl., Math. USSR-Sb. 50 (1985), 465-494. MR 85g:47041
  • [Pe2] V.V. PELLER, Hankel operators and multivariate stationary processes, Proc. Symp. Pure Math. 51 (1990), 357-371. MR 91m:47034
  • [Pe3] V.V. PELLER, Hankel operators and continuity properties of best approximation operators, Algebra i Analiz 2:1 (1990), 163-189; English transl. in Leningrad Math. J. 2 (1991), 139-160. MR 91e:41043
  • [Pe4] V.V. PELLER, Boundedness properties of the operators of best approximation by analytic and meromorphic functions, Ark. Mat. 30 (1992), 331-343. MR 95f:46038
  • [Pe5] V.V. PELLER, Approximation by analytic operator-valued functions, Harmonic analysis and operator theory (Caracas, 1994), 431-448, Contemp. Math., 189, Amer. Math. Soc., Providence, RI, 1995. MR 96i:47048
  • [PK] V.V. PELLER AND S.V. KHRUSCHEV, Hankel operators, best approximation and stationary Gaussian processes, Russian Math. Surveys 37 (1982), 53-124. MR 84e:47036
  • [PY1] V.V. PELLER AND N.J. YOUNG, Superoptimal analytic approximations of matrix functions, J. Functional Analysis 120 (1994), 300-343. MR 94m:47030
  • [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 97m:47031
  • [R] YU.A. ROZANOV, Stationary stochastic processes, Fizmatgiz, Moscow, 1963 (Russian). MR 28:2580
  • [S] I.B. SIMONENKO, Some general problems of the theory of the Riemann boundary-value problem, Izv. Akad. Nauk SSSR, Ser. Mat. 32 (1968), 1138-1146 (Russian). MR 38:3447
  • [To] V.A. TOLOKONNIKOV, Generalized Douglas algebras, Algebra i Analiz 3 (1991), 231-252 (Russian). MR 92m:46077
  • [Tr] S.R. TREIL, On superoptimal approximation by analytic and meromorphic matrix-valued functions, J. Functional Analysis 131 (1995), 386-414. MR 96g:47011
  • [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

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