The Products of Regularly Solvable Operators with Their Spectra in Direct Sum Spaces

Sobhy El-Sayed Ibrahim

Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt.

DOI: 10.4236/apm.2013.34060                

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].

Keywords

Product of Quasi-Differential Expressions; Regular and Singular Endpoints; Regularly Solvable Operators; Essential Spectra; Hilbert-Schmidt Integral Operators

Conflicts of Interest

The authors declare no conflicts of interest.

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