Representing non-weakly compact operators

González, Manuel ; Saksman, Eero ; Tylli, Hans-Olav

Studia Mathematica, Tome 113 (1995), p. 265-282 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $L^1$ and C(0,1), but R(L(E)/W(E)) identifies isometrically with the class of lattice regular operators on $ℓ^2$ for $E = ℓ^2(J)$ (here J is James’ space). Accordingly, there is an operator $T ∈ L(ℓ^2(J))$ such that R(T) is invertible but T fails to be invertible modulo $W(ℓ^2(J))$.

Détail

Publié le : 1995-01-01

Zbl 0832.47039

EUDML-ID : urn:eudml:doc:216174

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