FINITE RANK RIESZ OPERATORS

We provide conditions under which a Riesz operator defined on a Banach space is a finite rank operator. 2010 Mathematics Subject Classification. 47B06, 46L05. 1. Introduction. Let X be a Banach space, and denote by L(X) the Banach algebra of all bounded linear operators on X. An operator T ∈ L(X) is called a Riesz operator if the coset T + K(X) is quasinilpotent in the quotient algebra L(X)/K(X), where K(X) is the closed ideal of compact operators in L(X). We refer the reader to Dowson ((4), Part 2) for some basic properties of Riesz operators. For T ∈ L(X), denote the null space of T by N(T) and the range of T by R(T). The smallest integer n such that N(T n ) = N(T n+1 ) is called the ascent of T and it is denoted by α(T). The descent of T is the smallest integer n such that R(T n ) = R(T n+1 ) and it is denoted by δ(T). If M is a closed subspace of X invariant under T (i.e. T(M) ⊆ M), then the operator TM defined in L(X/M )b yTM(x + M) = Tx+ M is called the induced operator of T by M. The restriction of T to M is denoted by T|M.I fA is a C ∗ -algebra then the operator