Direct and inverse spectral problems for block Jacobi type bounded symmetric matrices related to the two dimensional real moment problem

Abstract

We generalize the connection between the classical power moment problem and the spectral theory of selfadjoint Jacobi matrices. In this article we propose an analog of Jacobi matrices related to some system of orthonormal polynomials with respect to the measure on the real plane. In our case we obtained two matrices that have a block three-diagonal structure and are symmetric operators acting in the space of $l_2$ type. With this connection we prove the one-to-one correspondence between such measures defined on the real plane and two block three-diagonal Jacobi type symmetric matrices. For the simplicity we investigate in this article only bounded symmetric operators. From the point of view of the two dimensional moment problem this restriction means that the measure in the moment representation (or the measure, connected with orthonormal polynomials) has compact support.

Key words: Classical and two dimensional moment problems, block three-diagonal matrix, eigenfunction expansion, generalized eigenvector, spectral problem.

Full Text

Article Information

Title	Direct and inverse spectral problems for block Jacobi type bounded symmetric matrices related to the two dimensional real moment problem
Source	Methods Funct. Anal. Topology, Vol. 20 (2014), no. 3, 219-251
MathSciNet	MR3242706
zbMATH	1324.47033
Milestones	Received 15/03/2014; Revised 25/06/2014
Copyright	The Author(s) 2014 (CC BY-SA)

Authors Information

Mykola E. Dudkin
National Technical University of Ukraine (KPI), 37 Peremogy Av., Kyiv, 03056, Ukraine

Valentyna I. Kozak
National Technical University of Ukraine (KPI), 37 Peremogy Av., Kyiv, 03056, Ukraine 

Citation Example

Mykola E. Dudkin and Valentyna I. Kozak, Direct and inverse spectral problems for block Jacobi type bounded symmetric matrices related to the two dimensional real moment problem, Methods Funct. Anal. Topology 20 (2014), no. 3, 219-251.

BibTex

@article {MFAT742,
    AUTHOR = {Dudkin, Mykola E. and Kozak, Valentyna I.},
     TITLE = {Direct and inverse spectral problems for block Jacobi type bounded symmetric matrices related to the two dimensional real moment problem},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {20},
      YEAR = {2014},
    NUMBER = {3},
     PAGES = {219-251},
      ISSN = {1029-3531},
  MRNUMBER = {MR3242706},
 ZBLNUMBER = {1324.47033},
       URL = {http://mfat.imath.kiev.ua/article/?id=742},
}