## Compact Jacobi matrices: from Stieltjes to Krein and M(a,b) (1996)

### BibTeX

@MISC{Assche96compactjacobi,

author = {Walter Van Assche},

title = {Compact Jacobi matrices: from Stieltjes to Krein and M(a,b)},

year = {1996}

}

### OpenURL

### Abstract

. --- In a note at the end of his paper Recherches sur les fractions continues, Stieltjes gave a necessary and sufficient condition when a continued fraction is represented by a meromorphic function. This result is related to the study of compact Jacobi matrices. We indicate how this notion was developped and used since Stieltjes, with special attention to the results by M. G. Krein. We also pay attention to the perturbation of a constant Jacobi matrix by a compact Jacobi matrix, work which basically started with Blumenthal in 1889 and which now is known as the theory for the class M (a; b). Mots-cles : Op'erateurs de Jacobi compacts, polynomes orthogonaux, perturbations compacts, th'eorie spectral Key-words: Compact Jacobi matrices, orthogonal polynomials, compact pertubations, spectral theory AMS Classification: 42C05 40A15 47A10 --- --- --- --- 1. A theorem of Stieltjes Stieltjes' research in Recherches sur les fractions continues [25] deals with continued fractions of the form...