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

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In this paper, we consider the general quasi-differential
expressions each of order *n* with complex coefficients and their formal
adjoints on the interval (*a*,*b*). It is shown in direct sum spaces 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 equation and its adjoint are in (the limit
circle case) that all well-posed extensions of the minimal operator have resolvents
which are HilbertSchmidt 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 extend those of
formally symmetric expression studied in
[1-10] and those of general quasi-differential expressions in [11-19].

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Advances in Pure Mathematics*, Vol. 3 No. 4, 2013, pp. 415-429. doi: 10.4236/apm.2013.34060.

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