Criterion for the Resolvent Set of Nonsymmetric Tridiagonal Operators

Abstract

Abstract. We study nonsymmetric tridiagonal operators acting in the Hubert space t1 and describe the spectrum and the resolvent set of such operators in terms of a continued fraction related to the resolvent. In this way we establish a connection between Padé approximants and spectral properties of nonsymmetric tridiagonal operators.

Generalized Jacobi operators in Krein spaces

“…2 Remark 5.4. In the case of nonsymmetric tridiagonal operators Theorem 5.3 was proved in [1]. In fact, we have extended the scheme proposed in [1] to the case of generalized Jacobi matrices.…”

“…Further, following the scheme proposed in [1], we find out a behavior of the associated polynomials at the points of the resolvent set. Proposition 7.5.…”

“…In fact, the scheme proposed in [1,7] (see also [29]) for investigation of the convergence of Padé approximants is generalized here to the nonregular case. The paper is organized as follows.…”

“…Weyl solutions of the three-term recurrence relations and Weyl functions are introduced in Section 4. In Section 5, following the scheme proposed in [1], the characterization of resolvent sets of generalized Jacobi operators is obtained. Section 6 is concerned with the case of periodic generalized Jacobi matrices.…”

Perturbation determinants and discrete spectra of semi-infinite non-self-adjoint Jacobi operators

“…The spectral theory of semi-infinite, self-adjoint Jacobi matrices is quite popular owing to their tight relation to the theory of orthogonal polynomials on the real line [18]. In contrast, there are only a few papers where semiinfinite, non-self-adjoint Jacobi matrices are examined [17,1,2,7,12,13].…”