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

Abstract

In this paper, we consider the general quasi-differential expressions \(\tau_{1}, \tau_{2}, \ldots, \tau_{n}\) each of order  with complex coefficients and their formal adjoints on the interval ( a, b ) . It is shown in direct sum spaces \(L_{w}^{2}\left(I_{p}\right), p=\) 1 , 2,..., N 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 product equation \(\left[\prod_{j=1}^{n} \tau_{j}-\lambda w\right] u=0\) and its adjoint  \(\left[\prod_{j=1}^{n} \tau_{j}^{+}-\bar{\lambda} w\right] v=\)  0 are in \(L_{w}^{2}(a, b)\)  (the limit circle case) that all well-posed extensions of the minimal operator T₀ (\(\tau_{1}, \tau_{2}, \ldots, \tau_{n}\) ) 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. These results are extension of those of formally symmetric expressions and those of general quasi-differential expressions.