Elementary operators, finite ascent, range closure and compactness

Given Banach space operators Ai,Bi∈B(X)Ai,Bi∈B(X), 1⩽i⩽21⩽i⩽2, let ΦAB∈B(B(X))ΦAB∈B(B(X)) denote the elementary operator ΦAB(X)=A1XB1−A2XB2ΦAB(X)=A1XB1−A2XB2. Then ΦABΦAB has finite ascent ⩽1 for a number of fairly general choices of the operators AiAi and BiBi. This information is applied to prove some necessary and sufficient conditions for the range of ΦABΦAB to be closed and in deciding conditions on the tuples (A1,A2)(A1,A2) and (B1,B2)(B1,B2) so that ΦABn(X) compact for some integer n⩾1n⩾1 and operator X   implies ΦAB(X)ΦAB(X) compact. This generalizes some well known results of Anderson and Foiaş [4], and Yosun [25]. Also considered is the question: What is a necessary and sufficient condition (on the tuples (A1,A2)(A1,A2), (B1,B2)(B1,B2) and ΦABΦAB) for ΦABn to be compact for some integer n⩾1n⩾1?