TITLE:
On the Domains of General Ordinary Differential Operators in the Direct Sum Spaces
AUTHORS:
Sobhy El-Sayed Ibrahim
KEYWORDS:
Quasi-Differential Expressions, Regular and Singular Equations, Minimal and Maximal Operators, Regularly Solvable Operators, J-Self-Adjoint Extension, Boundary Conditions
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.12 No.3,
March
22,
2022
ABSTRACT: Given general quasi-differential expressions , each of order n with complex coefficients and their formal adjoint are 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 expression in the direct sum Hilbert spaces . The domains of these operators are described in terms of boundary conditions involving -solutions of the equations and their adjoint on the intervals [ap,bp). 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 a special case, where J denotes complex conjugation.