TITLE:
The Products of Regularly Solvable Operators with Their Spectra in Direct Sum Spaces
AUTHORS:
Sobhy El-Sayed Ibrahim
KEYWORDS:
Product of Quasi-Differential Expressions; Regular and Singular Endpoints; Regularly Solvable Operators; Essential Spectra; Hilbert-Schmidt Integral Operators
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.3 No.4,
July
8,
2013
ABSTRACT: In this paper, we consider the general quasi-differential
expressions each of order n with complex coefficients and their formal
adjoints on the interval (a,b). It is shown in direct sum spaces of functions
defined on each of the separate intervals with the cases of one and two
singular end-points and 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 HilbertSchmidt 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
[1-10] and those of general quasi-differential expressions in [11-19].