Studies on the Domains of Regularly Solvable Operators in the Direct Sum Spaces

Abstract

Given a general quasi-differential expressions \(\tau\)₁ ,\(\tau\)₂ ,...,\(\tau\)_n each of order n with complex coefficients and their formal adjoint are \(\tau\)₁⁺ ,\(\tau\)₂⁺ ,...,\(\tau\)_n⁺ on the interval [a,b) respectively, we give a characterization of all regularly solvable operators and their adjoints generated by a general ordinary quasi-differential expressions \(\tau\)_jp  in the direct sum Hilbert spaces L²_w(a_p,b_p),p = 1,...,N. The domains of these operators are described in terms of boundary conditions involving L²_w(a_p,b_p)- solutions of the equations \(\tau\)_jp [y] = \(\lambda\) wy and its adjoint \(\tau\)⁺_jp[Z] = \(\lambda\) wy (\(\lambda\)\(\in\)\(\not\subset\)) on the intervals [a_p,b_p). This characterization is an extension of those obtained in the case of one interval with one and two singular end-points of the interval (a, b), and is a generalization of those proved in the case of self-adjoint and J- self-adjoint differential operators as special case, where J denotes complex conjugation.