# Representing non-weakly compact operators

Studia Mathematica (1995)

• Volume: 113, Issue: 3, page 265-282
• ISSN: 0039-3223

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## Abstract

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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 ${\ell }^{2}$ for $E={\ell }^{2}\left(J\right)$ (here J is James’ space). Accordingly, there is an operator $T\in L\left({\ell }^{2}\left(J\right)\right)$ such that R(T) is invertible but T fails to be invertible modulo $W\left({\ell }^{2}\left(J\right)\right)$.

## How to cite

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González, Manuel, Saksman, Eero, and Tylli, Hans-Olav. "Representing non-weakly compact operators." Studia Mathematica 113.3 (1995): 265-282. <http://eudml.org/doc/216174>.

@article{González1995,
abstract = {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 $W(ℓ^2(J))$.},
author = {González, Manuel, Saksman, Eero, Tylli, Hans-Olav},
journal = {Studia Mathematica},
keywords = {representation; weak Calkin algebra; weakly compact operators; lattice regular operators},
language = {eng},
number = {3},
pages = {265-282},
title = {Representing non-weakly compact operators},
url = {http://eudml.org/doc/216174},
volume = {113},
year = {1995},
}

TY - JOUR
AU - González, Manuel
AU - Saksman, Eero
AU - Tylli, Hans-Olav
TI - Representing non-weakly compact operators
JO - Studia Mathematica
PY - 1995
VL - 113
IS - 3
SP - 265
EP - 282
AB - 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 $W(ℓ^2(J))$.
LA - eng
KW - representation; weak Calkin algebra; weakly compact operators; lattice regular operators
UR - http://eudml.org/doc/216174
ER -

## References

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