1994/04/14 by Gonzalez, Manuel, Saksman, Eero, Tylli, H. · 1 citation
#47D #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.math/9404211
For each S ∈ L(E) (with E a Banach space) the operator R(S) ∈ L(E**/E) is defined by R(S)(x**+E) = S**x**+E (x**∈ E**). We study mapping properties of the correspondence S→ R(S), which provides a representation R of the weak Calkin algebra L(E)/W(E) (here W(E) denotes the weakly compact operators on E). Our results display strongly varying behaviour of R. For instance, there are no non--zero compact operators in Im(R) in the case of L1 and C(0,1), but R(L(E)/W(E)) identifies isometrically with the class of lattice regular operators on ℓ2 for E=ℓ2(J) (here J is the James' space). Accordingly, there is an operator T ∈ L(ℓ2(J)) such that R(T) is invertible but T fails to be invertible modulo W(ℓ2(J)).