# Representing non-weakly compact operators

Manuel González; Eero Saksman; Hans-Olav Tylli

Studia Mathematica (1995)

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

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topGonzález, 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 -

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