IJPAM: Volume 70, No. 5 (2011)

ON THE ESSENTIAL SPECTRA FOR PRODUCTS
OF THE GENERAL QUASI-DIFFERENTIAL
OPERATORS AND THEIR ADJOINTS

Sobhy El-Sayed Ibrahim
College of Basic Education
Department of Mathematics
P.O. Box 34053, Al-Edailiyah, 73251, KUWAIT


Abstract. The general quasidifferential expressions $ \tau _1,\tau _2,...,\tau _n $ each of order $ n $ with complex coefficients and their formal adjoints are considered on the interval $ (a,b)$. It is shown in the cases of one and two singular end-points when all solutions of the equation $ [\Pi _{j=1}^n\tau _j-\lambda w]u=,0 $ and its adjoint $ [\Pi _{j=1}^n\tau _j^{+}-\stackrel{\_}{\lambda }w]v= 0 $ are in $ L_w^2(a,b) $ (the limit circle case) that all well-posed extensions of the minimal operator $ T_0(\tau _1,\tau _2,...,\tau _n^{}) $ 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 $ \tau $ studied in [1], [15] and those of general quasidifferential expressions $ \tau $ in [10], [11], [13].

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
Year: 2011
Volume: 70
Issue: 5