ON THE ESSENTIAL SPECTRA OF GENERAL DIFFERENTIAL OPERATORS

In this paper, it is shown in the cases of one and two singular end-points and when all solutions of the equation $M[u]-\lambda uw=0$, and its adjoint $M^+[v] -\lambda wv = 0$ are in $L_w^2 (a, b)$ (the limit circle case) with $f\in L^2_w(a,b)$ for $M[u]-\lambda wu=wf$ that all well-posed extensions of the minimal operator $T_0(M)$ generated by a general ordinary quasi-differential expression $M$ of $n$-th order with complex coefficients have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly slovable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric expression $M$ studied in [1] and [12], and also extend those proved in [8] in the case of one singular end-point of the interval [a,b).