Criterion for the resolvent set of nonsymmetric tridiagonal operators

Authors:
A. I. Aptekarev, V. Kaliaguine and W. Van Assche

Journal:
Proc. Amer. Math. Soc. **123** (1995), 2423-2430

MSC:
Primary 47B37; Secondary 41A21, 47A05, 47B39

DOI:
https://doi.org/10.1090/S0002-9939-1995-1254830-5

MathSciNet review:
1254830

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study nonsymmetric tridiagonal operators acting in the Hilbert space 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.

**[1]**N. I. Akhiezer,*The classical moment problem*, Fizmatgiz, Moscow, 1961; English transl., Oliver and Boyd, Edinburgh.**[2]**N. I. Akhiezer and I. M. Glazman,*Theory of linear operators in Hilbert space*, Nauka, Moscow, 1966; English transl., Pitman, Boston.**[3]**A. I. Aptekarev,*Asymptotic properties of polynomials orthogonal on a system of contours and periodic motions of Toda lattices*, Mat. Sb.**125**(167) (1984), 231-258; English transl., Math. USSR-Sb.**53**(1986), 233-260. MR**764479 (86g:35166)****[4]**A. I. Aptekarev and E. M. Nikishin,*The scattering problem for a discrete Sturm-Liouville operator*, Mat. Sb.**121**(163) (1983), 327-358; English transl., Math. USSR-Sb.**49**(1984), 325-355. MR**708000 (85d:39004)****[5]**S. Demko, W. F. Moss, and P. W. Smith,*Decay rates for the inverse of band matrices*, Math. Comp.**43**(1984), 491-499. MR**758197 (85m:15002)****[6]**J.S. Geronimo and K.M. Case,*Scattering theory and polynomials orthogonal on the real line*, Trans. Amer. Math. Soc.**258**(1980), 467-494. MR**558185 (82c:81138)****[7]**J.S. Geronimo and W. Van Assche,*Orthogonal polynomials with asymptotically periodic recurrence coefficients*, J. Approx. Theory**46**(1986), 251-283. MR**840395 (87i:42035)****[8]**W.B. Jones and W.J. Thron,*Continued fractions: Analytic theory and applications*, Encyclopedia Math. Appl., vol. 11, Addison-Wesley, Reading, MA, and Cambridge Univ. Press, Cambridge, 1980. MR**595864 (82c:30001)****[9]**M.A. Krasnoselskiĭ and M.G. Krein,*Fundamental theorems on the extension of Hermitian operators and certain of their applications to the theory of orthogonal polynomials and the problem of moments*, Uspekhi Mat. Nauk**2(19)**(1947), no. 3, 60-106. (Russian) MR**0026759 (10:198b)****[10]**E.M. Nikishin,*Discrete Sturm-Liouville operators and some problems of function theory*, Trudy Sem. Petrovsk.**10**(1984), 3-77; English transl., J. Soviet Math.**35**(1986), 2679-2744. MR**778879 (86h:39007)****[11]**E.M. Nikishin and V.N. Sorokin,*Rational approximations and orthogonality*, Nauka, Moscow, 1988; English transl., Transl. Math. Monographs, vol. 92, Amer. Math. Soc., Providence, RI. MR**953788 (89m:30079)****[12]**W. Van Assche,*Asymptotics for orthogonal polynomials and three-term recurrences*, Orthogonal Polynomials: Theory and Practice (P. Nevai, ed.), NATO-Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 294, Kluwer Academic Publishers, Dordrecht, 1990, pp. 435-462. MR**1100305 (92k:42038)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47B37,
41A21,
47A05,
47B39

Retrieve articles in all journals with MSC: 47B37, 41A21, 47A05, 47B39

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1254830-5

Keywords:
Tridiagonal operators,
resolvents,
Padé approximation

Article copyright:
© Copyright 1995
American Mathematical Society