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1997 | 38 | 1 | 75-104
Tytuł artykułu

An operator-theoretic approach to truncated moment problems

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We survey recent developments in operator theory and moment problems, beginning with the study of quadratic hyponormality for unilateral weighted shifts, its connections with truncated Hamburger, Stieltjes, Hausdorff and Toeplitz moment problems, and the subsequent proof that polynomially hyponormal operators need not be subnormal. We present a general elementary approach to truncated moment problems in one or several real or complex variables, based on matrix positivity and extension. Together with the construction of a "functional calculus" for the columns of the associated moment matrix, our operator-theoretic approach allows us to obtain existence theorems for the truncated complex moment problem, in case the columns satisfy one of several natural constraints. We also include an application to the Riemann-quadrature problem from numerical analysis.
Słowa kluczowe
Rocznik
Tom
38
Numer
1
Strony
75-104
Opis fizyczny
Daty
wydano
1997
Twórcy
  • Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242, U.S.A.
Bibliografia
  • [Agl] J. Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985), 203-217.
  • [AK] N. I. Ahiezer and M. G. Krein, Some Questions in the Theory of Moments, Transl. Math. Monographs 2, Amer. Math. Soc., Providence, 1962.
  • [Akh] N. I. Akhiezer, The Classical Moment Problem, Hafner, New York, 1965.
  • [And] T. Ando, Truncated moment problems for operators, Acta Sci. Math. (Szeged) 31 (1970), 319-333.
  • [Atz] A. Atzmon, A moment problem for positive measures on the unit disc, Pacific J. Math. 59 (1975), 317-325.
  • [Ber] C. Berg, The multidimensional moment problem and semigroups, in: Moments in Mathematics, Proc. Sympos. Appl. Math. 37, Amer. Math. Soc., 1987, 110-124.
  • [BCJ] C. Berg, J. P. R. Christensen and C. U. Jensen, A remark on the multidimensional moment problem, Math. Ann. 223 (1979), 163-169.
  • [BeM] C. Berg and P. H. Maserick, Polynomially positive definite sequences, ibid. 259 (1982), 487-495.
  • [Bra] J. Bram, Subnormal operators, Duke Math. J. 22 (1955), 75-94.
  • [Cas] G. Cassier, Problème des moments sur un compact de $R^n$ et décomposition des polynômes à plusieurs variables, J. Funct. Anal. 58 (1984), 254-266.
  • [Con] J. B. Conway, Subnormal Operators, Pitman, London, 1981
  • [CoC] M. Cotlar and R. Cignoli, An Introduction to Functional Analysis, North-Holland, Amsterdam, 1974.
  • [Cu1] R. Curto, Quadratically hyponormal weighted shifts, Integral Equations Operator Theory 13 (1990), 49-66.
  • [Cu2] R. Curto, Joint hyponormality: A bridge between hyponormality and subnormality, in: Proc. Sympos. Pure Math. 51, Part 2, Amer. Math. Soc., 1990, 69-91.
  • [Cu3] R. Curto, Polynomially hyponormal operators on Hilbert space, Rev. Un. Mat. Argentina 37 (1991), 29-56.
  • [CuF1] R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations Operator Theory 17 (1993), 202-246.
  • [CuF2] R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, II, ibid. 18 (1994), 369-426.
  • [CuF3] R. Curto and L. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math. 17 (1991), 603-635.
  • [CuF4] R. Curto and L. Fialkow, Solution of the truncated complex moment problem for flat data, Mem. Amer. Math. Soc. 119 (1996).
  • [CuF5] R. Curto and L. Fialkow, Flat extensions of positive moment matrices: Relations in analytic or conjugate terms, in: Oper. Theory Adv. Appl., to appear.
  • [CuF6] R. Curto and L. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations, Mem. Amer. Math. Soc., to appear.
  • [CMX] R. Curto, P. Muhly and J. Xia, Hyponormal pairs of commuting operators, in: Oper. Theory Adv. Appl. 35, Birkhäuser, 1988, 1-22.
  • [CuP1] R. Curto and M. Putinar, Existence of non-subnormal polynomially hyponormal operators, Bull. Amer. Math. Soc. 25 (1991), 373-378.
  • [CuP2] R. Curto and M. Putinar, Nearly subnormal operators and moment problems, J. Funct. Anal. 115 (1993), 480-497.
  • [Fan] P. Fan, A note on hyponormal weighted shifts, Proc. Amer. Math. Soc. 92 (1984), 271-272.
  • [Fia] L. Fialkow, Positivity, extensions and the truncated complex moment problem, in: Contemp. Math. 185, Amer. Math. Soc., 1995, 133-150.
  • [Fug] B. Fuglede, The multidimensional moment problem, Exposition. Math. 1 (1983), 47-65.
  • [Hal] P. R. Halmos, Normal dilations and extensions of operators, Summa Brasil. Math. 2 (1950), 124-134.
  • [Hau] F. Hausdorff, Momentprobleme für ein endliches Intervall, Math. Z. 16 (1923), 220-248.
  • [Hav1] E. K. Haviland, On the momentum problem for distributions in more than one dimension, Amer. J. Math. 57 (1935), 562-568.
  • [Hav2] E. K. Haviland, On the momentum problem for distributions in more than one dimension, Part II, ibid. 58 (1936), 164-168.
  • [Hil] D. Hilbert, Über die Darstellung definiter Formen als Summen von Formenquadraten, Math. Ann. 32 (1888), 342-350.
  • [JeL] N. Jewell and A. Lubin, Commuting weighted shifts and analytic function theory in several variables, J. Operator Theory 1 (1979), 207-223.
  • [Jos1] A. Joshi, Hyponormal polynomials of monotone shifts, Ph.D. dissertation, Purdue University, 1971.
  • [Jos2] A. Joshi, Hyponormal polynomials of monotone shifts, Indian J. Pure Appl. Math. 6 (1975), 681-686.
  • [KrN] M. G. Krein and A. Nudel'man, The Markov Moment Problem and Extremal Problems, Transl. Math. Monographs 50, Amer. Math. Soc., Providence, 1977.
  • [Lan] H. Landau, Classical background of the moment problem, in: Moments in Mathema- tics, Proc. Sympos. Appl. Math. 37, Amer. Math. Soc., 1987, 1-15.
  • [Li] X. Li, On positive moment sequences, Ph.D. dissertation, Virginia Tech. Univ., 1993.
  • [McC] J. McCarthy, private communication.
  • [McCY] J. McCarthy and L. Yang, Subnormal operators and quadrature domains, preprint, 1995.
  • [McCP] S. McCullough and V. Paulsen, A note on joint hyponormality, Proc. Amer. Math. Soc. 107 (1989), 187-195.
  • [McG] J. L. McGregor, Solvability criteria for certain N-dimensional moment problems, J. Approx. Theory 30 (1980), 315-333.
  • [Mys] I. P. Mysovskikh, On Chakalov's Theorem, USSR Comp. Math. 15 (1975), 221-227.
  • [Nar] F. J. Narcowich, R-operators II. On the approximation of certain operator-valued analytic functions and the Hermitian moment problem, Indiana Univ. Math. J. 26 (1977), 483-513.
  • [Pru] B. Prunaru, Invariant subspaces for polynomially hyponormal operators, preprint, 1995.
  • [Pu1] M. Putinar, A two-dimensional moment problem, J. Funct. Anal. 80 (1988), 1-8.
  • [Pu2] M. Putinar, The L problem of moments in two dimensions, ibid. 94 (1990), 288-307.
  • [Pu3] M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J. 42 (1993), 969-984.
  • [Pu4] M. Putinar, Extremal solutions of the two-dimensional L-problem of moments, J. Funct. Anal. 136 (1996), 331-364.
  • [Pu5] M. Putinar, Quadrature domains and hyponormal operators, lecture at SEAM XI, Georgia Tech. Univ., Atlanta, 1995.
  • [Pu6] M. Putinar, On Tchakaloff's Theorem, preprint, 1995.
  • [Rez1] B. Reznick, Sums of even powers of real linear forms, Mem. Amer. Math. Soc. 463 (1992).
  • [Rez2] B. Reznick, e-mail communication.
  • [Sar] D. Sarason, Moment problems and operators on Hilbert space, in: Moments in Mathematics, Proc. Sympos. Appl. Math. 37, Amer. Math. Soc., 1987, 54-70.
  • [Sch1] K. Schmüdgen, An example of a positive polynomial which is not a sum of squares of polynomials. A positive, but not strongly positive functional, Math. Nachr. 88 (1979), 385-390.
  • [Sch2] K. Schmüdgen, The K-moment problem for semi-algebraic sets, Math. Ann. 289 (1991), 203-206.
  • [SeS] Z. Sebestyén and J. Stochel, Restrictions of positive self-adjoint operators, Acta Sci. Math. (Szeged) 55 (1991), 149-154.
  • [ShT] J. Shohat and J. Tamarkin, The Problem of Moments, Math. Surveys 1, Amer. Math. Soc., Providence, 1943.
  • [Smu] J. L. Smul'jan, An operator Hellinger integral, Mat. Sb. 91 (1959), 381-430 (in Russian).
  • [Sta] J. Stampfli, Which weighted shifts are subnormal, Pacific J. Math. 17 (1966), 367-379.
  • [StSz1] J. Stochel and F. H. Szafraniec, A characterization of subnormal operators, in: Spectral Theory of Linear Operators and Related Topics, Birkhäuser, 1984, 261-263.
  • [StSz2] J. Stochel and F. H. Szafraniec, Unbounded weighted shifts and subnormality, Integral Equations Operator Theory 12 (1989), 146-153.
  • [StSz3] J. Stochel and F. H. Szafraniec, On normal extensions of unbounded operators, III. Spectral properties, Publ. RIMS 25 (1989), 105-139.
  • [StSz4] J. Stochel and F. H. Szafraniec, Algebraic operators and moments on algebraic sets, Portugal. Math. 51 (1994), 1-21.
  • [Str] A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall, 1971.
  • [Sza1] F. H. Szafraniec, Boundedness of the shift operator related to positive definite forms: An application to moment problems, Ark. Mat. 19 (1981), 251-259.
  • [Sza2] F. H. Szafraniec, Moments on compact sets, in: Prediction Theory and Harmonic Analysis, V. Mandrekar and H. Salehi (eds.), North-Holland, Amsterdam, 1983, 379-385.
  • [Tch] V. Tchakaloff, Formules de cubatures mécaniques à coefficients non négatifs, Bull. Sci. Math. 81 (1957), 123-134.
  • [Wol] Wolfram Research, Inc., Mathematica, Version 2.1, Champaign, Illinois, 1992.
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Bibliografia
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