Finite Rank Riesz Operators

Abstract

We provide conditions under which a Riesz operator defined on a Banach space is a finite rank operator.

Riesz operators with finite rank iterates

“…Mathieu [11] generalised this result by proving that a weakly compact homomorphism defined on a C * -algebra with range in a normed algebra is a finite rank operator. More recently, Koumba and the second named author [7,Example 3.1] have given an example of a homomorphism defined on a C * -algebra that is a Riesz operator, but not a finite rank operator. However, if a homomorphism T defined on a C * -algebra is a Riesz operator with finite ascent n, then T n is a finite rank operator [7,Theorem 3.3].…”

“…More recently, Koumba and the second named author [7,Example 3.1] have given an example of a homomorphism defined on a C * -algebra that is a Riesz operator, but not a finite rank operator. However, if a homomorphism T defined on a C * -algebra is a Riesz operator with finite ascent n, then T n is a finite rank operator [7,Theorem 3.3]. In the present work, we seek similar results beyond the class of homomorphisms, that is, we consider Riesz operators defined on general Banach spaces.…”