On a spectral problem of Atkinson

Abstract

has a non-zero solution f~9l, ATKINSON proved the theorem that the zeros of the polynomial T(~) have no finite limit-point. In the case p = 1 this is a well-known fact proved in this general situation by F. RIusz [see 2 or :~] ; RtF~SZ' method, however, does not seem to apply for p > 1. This is not surprising as even for ordinary polynomials, to which our problem reduces if 9t is of dimension 1, the problem of the zeros lies much deeper for general p's than for p = 1. in this paper we give another proof of ATKINSON'S theorem, which, if not shorter, seems in some respect simpler than the original one. The main difference between the two methods is that we use spectral manifolds while ATKINSON uses eigen-manifolds, and that we apply the convenient tool of contour integration. Before going into the proof we recall, for sake of convenient reference, some known facts and definitions: For an arbitrary bounded (or at least closed) linear transformation T of the Banach space 'J~. into itself, the resolvent set e(T) is defined as the set