Search

AbstractWe initiate a study of structural properties of the quotient algebra ${\mathcal{K}}(X)/{\mathcal{A}}(X)$of the compact-by-approximable operators on Banach spaces $X$failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from $c_{0}$into ${\mathcal{K}}(Z)/{\mathcal{A}}(Z)$, where $Z$belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a nonseparable space $c_{0}(\unicode[STIX]{x1D6E4})$into ${\mathcal{K}}(Z_{FJ})/{\mathcal{A}}(Z_{FJ})$, where $Z_{FJ}$is a universal compact factorisation space arising from the work of Johnson and Figiel.

LetA=(A1,…,An) and (B1,…,Bn) ben-tuples of bounded linear operators on a Banach spaceE. The corresponding elementary operator EA,Bis the mapS↦∑ni=1AiSBionL(E), and Ea,bdenotes the induced operators↦∑ni=1aisbion the Calkin algebra C(E)=L(E)/K(E). Heret=T+K(E) forT∈L(E). We establish that ifEhas a 1-unconditional basis, thendist(Ea,b,W(C(E)))=‖Ea,b‖⩽dist (EA,B,W(L(E))),for all elementary operators EA,BonL(E), whereW(·) stands for the weakly compact operators. There is equality throughout ifE=ℓp, 1

Abstract.: The structure of the subspaces having the Dunford-Pettis property (DPP) is studied, where is the space of all compact operators on and . The following conditions are shown to be equivalent: (i) M has the DPP, (ii) M is isomorphic to a subspace of (iii) the sets and are relatively compact for all and . The equivalence between (i) and (iii) was recently proven in the case of arbitrary Hilbert spaces by Brown and lger. It is also shown that (i) and (ii) are equivalent for subspaces . This result is optimal in the sense that for there is a DPP-subspace that fails to be isomorphic to a subspace of .