On the product of singular differential operators with their essential spectra in direct sum space

Abstract

We consider the product of general quasi-differential expressions $\tau_{1},\tau _{2},...,\tau _{n}$ each of order $n$ with complex\textbf{\ }%coefficients and their formal adjoints$\;$on the interval $(a,b).$ It is shown in direct sum spaces $L_{w}^{2}(I_{p}),\;p=1,\;...,N\;$of functions defined on each of the separate intervals with the case of one singular end-points and when all solutions of the equation $[\Pi _{j=1}^{n}\tau_{j}-\lambda w]u=0$ and its adjoint $[\Pi _{j=1}^{n}\tau _{j}^{+}-\stackrel{%
\_}{\lambda }w]v=$ $0$\ are in $L_{w}^{2}(a,b)$ (the limit circle case) that all well-posed extensions of the minimal operator $T_{0}(\tau _{1},\tau _{2},\;...,\tau _{n}^{{}})$ have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. Thisimplies that all the regularly solvable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric differential expression $\tau $ studied in [1,12,13,14,18,20] and those of general quasi-differential expression $\tau $ in [3,6,7,9,11,15,16].