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Extremal solutions of the two-dimensional \(L\)-problem of moments. II. (English) Zbl 0910.47011
This article is the continuation of the first article with the same title of the same author [published in J. Funct. Anal. 136, No. 2, 331-364 (1996)]. The purpose of this article is to apply the techniques developed in the first article to all extremal solutions of the \(L\)-problem.
We give the following definitions to introduce some of the important results. Let \(K\) be a compact subset of the complex plane, let \(L\) be a fixed positive constant, and let \(N\) be a fixed positive integer. We are interested in classifying and characterizing in intrinsic terms the moments \(a_{nm}= \int_K\varphi(x,y) x^ny^mdA\), \(0\leq m\leq m+ n\leq N\), of a measurable function \(\varphi\) on \(K\) which satisfies \(0\leq\varphi\leq L\), \(dA\)-a.e., where \(dA\) is the planar Lebesgue measure. We denote \(\sum\) the collection of all vectors \(a= (a_{kl})_{k+l\leq N}\in\mathbb{R}^d\), \(d={(N+1)(N+2)\over 2}\) which arise as the moments of a function \(\varphi\). The author follows methods of M. G. Krein’s convexity theory.
In this paper, Section 1 is an introduction. Section 2 reminds the main results of M. G. Krein which characterize the extremal solutions of the truncated \(L\)-problem of moments. In Section 3, the research of the associated hyponormal operators is presented and will be established via a rational cyclicity criterion, the existence of a general analytic model of them, valid for all extremal solutions described in Section 2. In Section 4, the canonical exponential kernel of an extremal solution of the momentum problem is analyzed.
The author presents an intrinsic characterization of all kernels and interprets the \(L\)-problem of moments as an interpolation problem of the Carathéodory-Fejér type, for a class of analytic functions defined in the bidisk.

47A57 Linear operator methods in interpolation, moment and extension problems
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[1] Carey, R.W.; Pincus, J.D., An exponential formula for determining functions, Indiana univ. math. J., 23, 1031-1042, (1974) · Zbl 0288.47020
[2] Clancey, K., The Cauchy transform of the principal function associated with an nonnormal operator, Indiana univ. math. J., 34, 21-32, (1985) · Zbl 0592.47012
[3] Clancey, K., Hilbert space operators with one dimensional self-commutators, J. operator theory, 13, 265-289, (1985) · Zbl 0588.47043
[4] Davis, P., The Schwarz function and its applications, Carus math. monographs, 17, (1974), Math. Assoc. of America
[5] de Branges, L., Hilbert spaces of entire functions, (1968), Prentice-Hall · Zbl 0157.43301
[6] Foiaş, C.; Frazho, A.E., The commutant lifting approach to interpolation problems, (1990), Birkhäuser-Verlag Basel · Zbl 0718.47010
[7] B. Gustafsson, M. Putinar, An exponential transform and regularity of free boundaries in two dimensions, Ann. Sci. Norm. Sup. Pisa · Zbl 0937.32007
[8] Karlin, S.; Studden, W.J., Tchebycheff systems, with applications in analysis and statistics, (1966), Interscience New York · Zbl 0153.38902
[9] Krein, M.G., The ideas of P. L. Chebyshev and A. A. Markov in the theory of limiting values of integrals and their further developments, Uspekhi mat. nauk., 6, 3-120, (1951)
[10] Krein, M.G.; Nudelman, A.A., Markov moment problem and extremal problems, Translations, American math. society, 50, (1977), Amer. Math. Soc Providence
[11] M. S. Livšic, Commuting Nonselfadjoint Operators and Collective Motions of Systems, Lecture Notes in Math. 1272, 4, 38, Springer-Verlag, Berlin
[12] Martin, M.; Putinar, M., Lectures on hyponormal operators, (1989), Birkhäuser Basel · Zbl 0684.47018
[13] Pincus, J.D.; Rovnyak, J., A representation for determining functions, Proc. amer. math. soc., 22, 498-502, (1969) · Zbl 0188.42801
[14] Pincus, J.D.; Xia, D.; Xia, J., An analytic model for operators with one-dimensional self-commutator, Int. equations operator theory, 7, 516-535, (1984) · Zbl 0592.47013
[15] Putinar, M., Extremal solutions of the two-dimensionalL, J. funct. anal., 136, 331-364, (1996) · Zbl 0917.47014
[16] Putinar, M., Linear analysis of quadrature domains, Ark. mat., 33, 357-376, (1995) · Zbl 0892.47025
[17] Sakai, M., Regularity of a boundary having a Schwarz function, Acta math., 166, 263-297, (1991) · Zbl 0728.30007
[18] Shapiro, H.S., The Schwarz function and its generalization to higher dimensions, Univ. of arkansas lecture notes in math., 9, (1992), Wiley New York
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