Representing non–weakly compact operators
نویسندگان
چکیده
For each S ∈ L(E) (with E a Banach space) the operator R(S) ∈ L(E/E) is defined by R(S)(x + E) = Sx + 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 L and C(0, 1), but R(L(E)/W (E)) identifies isometrically with the class of lattice regular operators on l for E = l(J) (here J is the James’ space). Accordingly, there is an operator T ∈ L(l(J)) such that R(T ) is invertible but T fails to be invertible modulo W (l(J)).
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