Abstract
Let T be a self-adjoint tridiagonal operator in a Hilbert space H with the orthonormal basis {e n}n=1∞, σ(T) be the spectrum of T and Λ(T) be the set of all the limit points of eigenvalues of the truncated operator T N. We give sufficient conditions such that the spectrum of T is discrete and σ(T) = Λ(T) and we connect this problem with an old problem in analysis.
[1] G.D. Alben, C.K. Chui, W.R. Madych, F.J. Narcowich and P.W. Smith: “Pade approximation of Stieltjes series”, J. Appr. Theory, Vol. 14, (1975), pp. 302–316. http://dx.doi.org/10.1016/0021-9045(75)90077-510.1016/0021-9045(75)90077-5Search in Google Scholar
[2] P. Deliyiannis and E.K. Ifantis: “Spectral theory of the difference equation f(n + 1) + f(n −1) = (E − φ(n))f (n)”, J. Math. Phys., Vol. 10, (1969), pp. 421–425. http://dx.doi.org/10.1063/1.166485510.1063/1.1664855Search in Google Scholar
[3] P. Hartman and A. Winter: “Separation theorems for bounded hermitian forms”, Amer. J. Math, Vol. 71, (1949), pp. 856–878. Search in Google Scholar
[4] E.K. Ifantis and P.D. Siafarikas: “An alternative proof of a theorem of Stieltjes and related results”, J. Comp. Appl. Math., Vol. 65, (1995), pp. 165–172. http://dx.doi.org/10.1016/0377-0427(95)00123-910.1016/0377-0427(95)00123-9Search in Google Scholar
[5] E.K. Ifantis and P. Panagopoulos: “Limit points of eigenvalues of truncated tridiagonal operators”, J. Comp. Appl. Math., Vol. 133, (2001), pp. 413–422. http://dx.doi.org/10.1016/S0377-0427(00)00663-410.1016/S0377-0427(00)00663-4Search in Google Scholar
[6] J. Rappaz: “Approximation of the spectrum of non compact operators given by the magnetohydrodynamic stability of plasma”, Numer. Math., Vol. 28, (1977), pp. 15–24. http://dx.doi.org/10.1007/BF0140385410.1007/BF01403854Search in Google Scholar
[7] T.J. Stieltjes: “Recherches sur les fractions continues”, Ann. Fac. Sci. Toulouse Mat., Vol. 8, (1894), J1–J122; Vol. 9, (1895), A1–A47; Oeuvres, Vol. 2, (1918), pp. 398–506. Search in Google Scholar
[8] M.H. Stone: “Linear Transformations in Hilbert space and their Applications to Analysis”, In: Amer. Math. Soc. Colloq. Publ., Vol. 15, Amer. Math. Soc., Providence, R.I. New York, 1932. Search in Google Scholar
[9] H.S. Wall: “On continued fractions which represent meromorphic functions”, Bull. Amer. Math. Soc., Vol. 39, (1933), pp. 946–952. http://dx.doi.org/10.1090/S0002-9904-1933-05778-610.1090/S0002-9904-1933-05778-6Search in Google Scholar
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