Completely continuous inverses of ordinary differential operators

Abstract

Many important properties, such as the absence of essential spectrum, follow when an ordinary differential operator comes from a differential expression such that the minimal operator has completely continuous inverse. This always happens in the case of a compact interval. On infinite intervals it happens much less frequently, however. To assure that the minimal operator is 1-1, we analyze the question only on intervals of type 1= [a, oo). By an ordinary differential expression r we mean an expression of type EOakDk, where ak,EzCk(I), and an is nonvanishing on I. With 1 <p< oo and 1 <q< oo, we define the minimal operator To,p,q in the usual fashion. (See Goldberg's book for details.) Our main result shows roughly that for formally selfadjoint differential expressions, To,2,2 has compact inverse if and only if the spectra of its selfadjoint extensions are completely dependent on the boundary conditions. A weaker result holds for To,p,p with p$2.