The Apostol–Fialkow Formula for Elementary Operators on Banach Spaces

LetA=(A1,…,An) and (B1,…,Bn) ben-tuples of bounded linear operators on a Banach spaceE. The corresponding elementary operator EA,Bis the mapS↦∑ni=1AiSBionL(E), and Ea,bdenotes the induced operators↦∑ni=1aisbion the Calkin algebra C(E)=L(E)/K(E). Heret=T+K(E) forT∈L(E). We establish that ifEhas a 1-unconditional basis, thendist(Ea,b,W(C(E)))=‖Ea,b‖⩽dist (EA,B,W(L(E))),for all elementary operators EA,BonL(E), whereW(·) stands for the weakly compact operators. There is equality throughout ifE=ℓp, 1