The Well-Posed Operators with Their Spectra in <i><span>L</span>^p<sub style="margin-left:-10px;">w</sub></i>-Spaces

Abstract

In this paper, we have considered the general ordinary quasi-differential operators generated by a general quasi-differential expression τ p,q in L p w -spaces of order n with complex coefficients and its formal adjoint τ + q',p' in L p w -spaces for arbitrary p , q ∈[1,∞) . We have proved in the case of one singular end-point that all well-posed extensions of the minimal operator T 0 ( τ p,q ) generated by such expression τ p,q and their formal adjoint on the interval [ a , b ) with maximal deficiency indices 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. Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions can be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while others are new.