Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 18 (2023), 73 -- 82

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This work is licensed under a Creative Commons Attribution 4.0 International License.

ON BOUNDED COMPLEX JACOBI MATRICES AND RELATED MOMENT PROBLEMS IN THE COMPLEX PLANE

Sergey M. Zagorodnyuk

Abstract. In this paper we consider the following moment problem: find a positive Borel measure μ on ℂ subject to conditions ∫ zn dμ = sn, n∈ℤ+, where sn are prescribed complex numbers (moments). This moment problem may be viewed (informally) as an extension of the Stieltjes and Hamburger moment problems to the complex plane. A criterion for the moment problem for the existence of a compactly supported solution is given. In particular, such moment problems appear naturally in the domain of complex Jacobi matrices. For every bounded complex Jacobi matrix its associated functional S has the following integral representation: S(p) = ∫ℂ p(z) dμ, with a positive Borel measure μ in the complex plane. An interrelation of the associated to the complex Jacobi matrix operator A0, acting in l2 on finitely supported vectors, and the multiplication by z operator in L2μ is discussed.

2020 Mathematics Subject Classification: 44A60
Keywords: Complex Jacobi matrix; moment problem; orthogonal polynomials; linear functional

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References

  1. N.I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965. MR0184042. Zbl 0135.33803.

  2. B. Beckermann, Complex Jacobi matrices, Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials. J. Comput. Appl. Math. 127 no. 1-2 (2001), 17-65. MR1808568. Zbl 0977.30007.

  3. B. Beckermann, M. Castro Smirnova, On the determinacy of complex Jacobi matrices, Math. Scand. 95 no. 2 (2004), 285-298. MR1808568. Zbl 1070.47018.

  4. Ju.M. Berezanskii, Expansions in eigenfunctions of selfadjoint operators, Amer. Math. Soc., Providence, RI, 1968. MR0222718. Zbl 0157.16601.

  5. J.S. Christiansen, M. Zinchenko, Lieb-Thirring inequalities for complex finite gap Jacobi matrices, Lett. Math. Phys. 107 no. 9 (2017), 1769-1780. MR3687263. Zbl 06793826.

  6. S.R. Garcia, E. Prodan, M. Putinar, Mathematical and physical aspects of complex symmetric operators, J. Phys. A 47 no. 35 (2014), 353001, 54 pp. MR3254868. Zbl 1338.81200.

  7. L.B. Golinskii, Perturbation determinants and discrete spectra of semi-infinite non-self-adjoint Jacobi operators, J. Spectr. Theory 12 no. 2 (2022), 835-856. MR4487492. Zbl 07597915.

  8. G.S. Guseinov. Determination of an infinite non-self-adjoint Jacobi matrix from its generalized spectral function, Mathematical Notes of the Academy of Sciences of the USSR 23 (1978) 130-136. MR0485911. Zbl 0428.47017.

  9. Y. Ikebe, N. Asai, Y. Miyazaki, D. Cai, The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application, Proceedings of the Fourth Conference of the International Linear Algebra Society (Rotterdam, 1994), Linear Algebra Appl. 241/243 (1996), 599-618. MR1400455. Zbl 0857.15001.

  10. G.V. Milovanovic, A.S. Cvetkovic, Complex Jacobi matrices and quadrature rules, Filomat No. 17 (2003), 117-134. MR2077354. Zbl 1056.65023.

  11. B.Sz.-Nagy, C. Foias, H. Bercovici, L. Kérchy, Harmonic analysis of operators on Hilbert space, Second edition, Revised and enlarged edition. Universitext. Springer, New York, 2010. MR2760647. Zbl 1234.47001.

  12. F. ℑch, The characteristic function for complex doubly infinite Jacobi matrices, Integral Equations Operator Theory 88 no. 4 (2017), 501-534. MR3694621. Zbl 1466.47022.

  13. G. Swiderski, Spectral properties of some complex Jacobi matrices, Integral Equations Operator Theory 92 no. 2 (2020), Paper No. 11, 24 pp. MR4070763. Zbl 1441.47039.

  14. B. Szőkefalvi-Nagy, C. Foias, H. Bercovici, L. Kérchy, Harmonic analysis of operators on Hilbert space, Second edition, Revised and enlarged edition. Universitext. Springer, New York, 2010. MR2760647. Zbl 1234.47001.

  15. H.S. Wall, Analytic theory of continued fractions, D. Van Nostrand Company, Inc., New York, N. Y., 1948. MR0025596. Zbl 0035.03601.

  16. S.M. Zagorodnyuk, Direct and inverse spectral problems for (2N+1)-diagonal, complex, symmetric, non-Hermitian matrices, Serdica Math. J. 30, No. 4 (2004), 471-482. MR2110489. Zbl 1081.39504.

  17. S.M. Zagorodnyuk, On the complex symmetric and skew-symmetric operators with a simple spectrum, SIGMA Symmetry Integrability Geom. Methods Appl. 7, (2011), Paper 016, 9 pp. MR2804580. Zbl 1218.44002.

  18. S. Zagorodnyuk, On the truncated multidimensional moment problems in ℂn, Axioms 11, no. 1 (2022), 20.

  19. S.M. Zagorodnyuk, On the similarity of complex symmetric operators to perturbations of restrictions of normal operators, Operators and Matrices 16, no. 1 (2022), 101-112. MR4428600. Zbl 1507.47060.

  20. S. Zagorodnyuk, Pencils of semi-infinite matrices and orthogonal polynomials Chapter in "Matrix Theory - Classics and Advances", Edited by Mykhaylo Andriychuk, by IntechOpen, London, United Kingdom, 2023.



Sergey M. Zagorodnyuk
V. N. Karazin Kharkiv National University,
School of Mathematics and Computer Sciences,
Department of Higher Mathematics and Informatics,
Svobody Square 4, 61022, Kharkiv, Ukraine.
e-mail: Sergey.M.Zagorodnyuk@gmail.com; zagorodnyuk@karazin.ua


http://www.utgjiu.ro/math/sma