The Apostol–Fialkow Formula for Elementary Operators on Banach Spaces

Abstract

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<p<∞. Our results extend and improve a corresponding structural result of Apostol and Fialkow (Canad. J. Math.38(1986), 1485–1524), which they proved forE=ℓ2using the non-commutative Weyl–von Neumann theorem due to Voiculescu. By contrast, our arguments are based on subsequence techniques from Banach space theory. As a byproduct we obtain a positive answer to the generalized Fong–Sourour conjecture for a large class of Banach spaces. We also explicitly compute the norm of the generalized derivations↦as−sbon C(ℓ2) (this improves a result due to Fong) and show that the resulting formula fails to hold on C(ℓp).