Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights.
type: Item
instance of: scholarly article
title: Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights.
zbMATH Open document ID: 1048.47019
DOI: 10.1007/BF01212702
author: Jan Janas
author: Marcin Moszyński
published in: Integral Equations and Operator Theory
publication date: 2002-01-01T00:00:00Z
review text: Jacobi matrices are defined as linear operators on \(\ell^2(\{1,2,\dots\})\) on the maximal domain of definition whose matrix \([J_{m,n}]^\infty_{m,n=1}\) with respect to the standard basis of \(\ell^2(\{1,2,\dots\})\) is tridiagonal and conjugate transpose: \(J_{m,m}= q_m\) are real numbers, \(J_{m,m+1}= \lambda_m= \overline{J_{m+1,m}}\), \(J_{m,n}= 0\) if \(| m- n|> 1\). It is assumed in the present paper that \(\lambda_m> 0\) for all \(m\). Jacobi matrices are symmetric operators (on a minimal domain), but are not necessarily selfadjoint. Various sufficient conditions are given for Jacobi matrices to be absolutely continuous selfadjoint operators, and to be selfadjoint operators with no point spectrum. One type of such conditions is derived by using a commutator result -- a generalization of the Putnam lemma [see \textit{C. R. Putnam}, ``Commutation properties of Hilbert space operators'' (Springer, New York) (1967; Zbl 0149.35104)] to unbounded operators. Another type of such conditions is given under monotonicity or near monotonicity of the sequence of weights \([\lambda_m]^\infty_{m=1}\). Several examples are given, and a comparison with previously known results is made.
reviewed by: Leiba Rodman
Mathematics Subject Classification ID: 47B36
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 1890588
zbMATH Keywords: Jacobi matrices
zbMATH Keywords: absolute continuity
zbMATH Keywords: selfadjointness
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1890588