The higher order differential operators in direct sum spaces

Abstract

W. N. Everitt and A. Zettl [l] introduced the concept of differential operator into the direct sum space, and provided an operator theoretic framework for the study of two generalized Sturm-Liouville differential operators together: M, defined on an interval I, and M, defined on I,. In particular, they characterized self-adjoint extensions of the minimal operator in the direct sum space in terms of boundary conditions. The aim of this paper is to study the same problem for the higher order differential operators in the direct sum spaces. Since Sturm-Liouville differential operators always have equal deficiency indices, the minimal operators in the direct sum spaces also have equal deficiency indices and self-adjoint extensions. But for higher order differential operators on an interval Z, as is well known, the values of the deficiency index are very complicated (e.g., see [7, Chap. 61). Hence the minimal operator in the direct sum space has equal deficiency indices only when the deficiency indices of two higher order differential operators satisfy certain conditions. In this paper, we use Cao’s and Sun’s methods [2, 31 to give conditions which characterize the domains of self-adjoint extensions of minimal operators.