The structure and asymptotic behavior of polynomially compact operators

Abstract

A. R. Bernstein and A. Robinson proved that every polynomially compact operator in Hilbert space has nontrivial invariant subspaces. This paper gives a strncture theorem for these operators. We show that a polynomially compact operator is the finite sum of translates of operators which have the property that a finiite power of the operator is compact. Furthermore, the spectrum of polynomially compact operators is completely de- scribed. Conditions are given to determine the weak and strong asymptotic behavior of a polynomially compact contraction in Hilbert space. We say that a bounded operator A on a complex Banach space B is polynomially compact if there is a nonzero complex polynomial p(z) such that the operator p(A) is compact. In ?1 we shall give a struc- ture theorem for these operators which reduces their study to the study of power compact operators. We shall show that a polynomially compact operator is just the finite sum of translates of operators which have the property that a finite power of the operator is com- pact. In case the space is a Hilbert space and the operator is a normal operator, then it is just the direct sum of translates of compact nor- mal operators and hence its complete structure can be determined. Furthermore, we can exactly describe the spectrum of these opera- tors. Our structure theorems, besides describing the spectrum of such operators, will have several applications which we present in ?2. It is immediate from the structure theorem that the existence of nontrivial invariant subspaces follows directly from the theorem in