Simplicity of linear ordinary differential operators

Abstract

Let A be a closed symmetric operator in a Hilbert space 6. A is said to be simple if there is no nontrivial reducing subspace in which A is self-adjoint. Note that if A is simple, then A does not have eigenvalues. In this article it is shown that the minimal closed symmetric operator T,, determined by a formally self-adjoint linear ordinary differential operator L on an interval (a, b) is simple if L is regular or quasi regular at one end. The proof uses a generalized resolvent and spectral function of T,, and the Stieltjes inversion formula. It is also shown that if L has order n = 1 or if L has real coefficients and order n = 2 (i.e., L is a Sturm-Liouville operator), operator), then T,, is simple or self-adjoint whatever the behavior of L at a and 6. A is said to be regular if every complex number 1 is a point of regular type, i.e., (A ZE)-l exists and is bounded. (Here E is the identity operator.) If A is regular, then A is simple. For, suppose A = A, @ A, , where A, is a nontrivial self-adjoint operator, and A, is closed symmetric. Then I is a point of regular type for A if and only if 1 is a point of regular type for A, and A, . But, since A, is self-adjoint, its spectrum is nonempty and therefore not every point is of regular type for A, . Hence, not every point is of regular type for A, and A is not regular. Thus, if A is regular, A is simple. The converse of this statement is not true. In Section 3, Remark 4, we shall give an example of a closed symmetric operator A which is simple but not regular. We note that the symmetric operators defined by Jacobi matrices and considered by Stone [8, Theorem IO.411 are simple whenever the deficiency mdex is (I, 1). This is because such a symmetric operator A has two different