IJPAM: Volume 70, No. 5 (2011)
OF THE GENERAL QUASI-DIFFERENTIAL
OPERATORS AND THEIR ADJOINTS
College of Basic Education
Department of Mathematics
P.O. Box 34053, Al-Edailiyah, 73251, KUWAIT
Abstract. The general quasidifferential expressions each of order with complex coefficients and their formal adjoints are considered on the interval . It is shown in the cases of one and two singular end-points when all solutions of the equation and its adjoint are in (the limit circle case) that all well-posed extensions of the minimal operator have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric expression studied in ,  and those of general quasidifferential expressions in , , .
Received: February 2, 2011
AMS Subject Classification: 34B05, 34B24, 47A10, 47E05
Key Words and Phrases: quasidifferential operators, regular and singular endpoints, regularly solvable operators, essential spectra, Hilbert-Schmidt integral operators
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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395