On the spectra of non-selfadjoint differential operators and their adjoints in direct sum spaces

The general ordinary quasidifferential expressionMpofnth order, with complex coefficients and its formal adjointMp+on any finite number of intervalsIp=(ap,bp),p=1,…,N, are considered in the setting of the direct sums ofLwp2(ap,bp)-spaces of functions defined on each of the separate intervals. And a number of results concerning the location of the point spectra and regularity fields of general differential operators generated by such expressions are obtained.

The spectra of well-posed operators

In this paper, the general ordinary quasidifferential expression M of nth order, with complex coefficients, and its formal adjoint M− are considered. It is shown in the case of two singular endpomts and when all solutions of the equation and the adjoint equation are in (the limit-circle case) that all well-posed extensions of the minimal operator T0(M) 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 of the standard essential spectra to be empty. These results extend those for the formally symmetric expression M studied in [1] and [14], and also extend those proved in [8] for one singular endpoint.