Simplicity of differential operators on an infinite interval

Abstract

A closed symmetric operator A in a Hilbert space !f~ is called simple if there is no nontrivial reducing subspace on which A is self-adjoint. The author [2] showed that if a formally self-adjoint ordinary differential operator L on an interval I is regular or quasiregular at one end of I, then the minimal closed symmetric operator T,, defined by L in $ = P(1) is simple. It follows that if L has order n = 1 or if L has order n ..= 2 and real coefficients, then T, is simple or self-adjoint. This is because in these cases either T, is self-adjoint or else L is regular or quasiregular at one end of 1. It is the purpose of this article to show that if 11 > 2, in particular, if n = 4, then T,, can be nonself-adjoint and not simple. This will be done by adducing an example of an I; on (-cc, co) for which T, is the same as the closed symmetric operator defined by an infinite matrix. From this infinite matrix one can determine the structure of T,, . In particular, T0 turns out to have an eigenvalue and is, therefore, not simple. Since T,, has deficiency index (2, 2), T, is also not self-adjoint. For notation and definitions we refer the reader to [2]. We shall make use of the following lemma.