Representing non-weakly compact operators
Studia Mathematica, Tome 113 (1995) no. 3, pp. 265-282
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
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 $W(ℓ^2(J))$.
@article{10_4064_sm_113_3_265_282,
author = {Manuel Gonz\'alez},
title = {Representing non-weakly compact operators},
journal = {Studia Mathematica},
pages = {265--282},
publisher = {mathdoc},
volume = {113},
number = {3},
year = {1995},
doi = {10.4064/sm-113-3-265-282},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm-113-3-265-282/}
}
Manuel González. Representing non-weakly compact operators. Studia Mathematica, Tome 113 (1995) no. 3, pp. 265-282. doi: 10.4064/sm-113-3-265-282
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