Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights.
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.
scientific article; zbMATH DE number 1890588