In [5] Kalf obtained a characterization of the Friedrichs extension TF of a general semi-bounded Sturm–Liouville operator T, the only assumptions made on the coefficients being those necessary for T to be defined. The domain D(TF) of TF was described in terms of ‘weighted’ Dirichiet integrals involving the principal and non-principal solutions of an associated non-oscillatory Sturm–Liouville equation. Conditions which ensure that members of D(TF) have a finite Dirichlet integral were subsequently given by Rosenberger in [7].

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Brown & Evans proposed a criterion for the validity of their series analogue of Everitt's extension of the Hardy-Littlewood inequality presupposes that the associated difference equation is in the strong-limit point at infinity. In this paper, we investigate the case when the difference equation involved is in the limit-circle case and non-oscillatory and also when the interval is finite.

SynopsisIn this article the expression τφ: = pφ + qφ with complex-valued coefficients is considered. We are particularly concerned with this expression when it is not formally symmetric, i.e., τ ≠ τ+, where τ+ is the formal adjoint of τ, and especially with the operators which are regularly solvable with respect to the minimal operators generated by τ and τ+ in the sense of W. D. Evans in [3]. This article is divided into five sections: Section 1 is an introduction, Section 2 is a brief study of the regular problem, in Section 3, some preliminary results in the singular case are displayed in Section 4, the joint field of regularity in the singular case is investigated and in Section 5, we discuss the case when .