The Alternative Dunford–Pettis Property for Subspaces of the Compact Operators

Abstract

Abstract.A Banach space X has the alternative Dunford-Pettis property if for every weakly convergent sequences (xn) → x in X and (x * n ) → 0 in X * with xn = x = 1 we have (x * n (xn)) → 0. We get a characterization of certain operator spaces having the alternative Dunford-Pettis property. As a consequence of this result, if H is a Hilbert space we show that a closed subspace M of the compact operators on H has the alternative Dunford-Pettis property if, and only if, for any h ∈ H, the evaluation operators from M to H given by S → Sh, S → S t h are DP1 operators, that is, they apply weakly convergent sequences in the unit sphere whose limits are also in the unit sphere into norm convergent sequences. We also prove a characterization of certain closed subalgebras of K(H) having the alternative Dunford-Pettis property by assuming that the multiplication operators are DP1.

The Alternative Dunford-Pettis Property in Subspaces of Operator Ideals

“…Here, we will show that similar consequences of [14], that extend some results of [1], remain valid for the DP1 property and a suitable class of closed subspaces of some operator ideals, where in this case the evaluation operators must be assumed DP1 operators.…”

“…Here we use similar techniques to those in [1] and [14] to obtain some characterizations of the DP1 property for suitable closed subspaces of some compact operator ideals between Banach spaces that extend some results of [1]. We need some notations.…”

“…Since P V and P W are finite rank operators, it is easy to check that the operators K → KP V and K → P W K from M into K w * (X * , Y ) are DP1 (see for instance, Remark 2.3 of [1]). Thus by hypothesis on (K n ) we have…”

“…Also, in [1], the authors study the DP1 property for closed subspaces of K(X, Y ), where X and Y admit Schauder basis and the basis of X is shrinking; and they proved some necessary and sufficient conditions for the DP1 property of suitable subspaces of K(X, Y ).…”

Dunford–Pettis properties, Hilbert spaces and projective tensor products

“…The DP1, the study of which was initiated in [28], is strictly weaker than the DP as exampled by infinite-dimensional Hilbert spaces and dual spaces of compact operators. Subsequent investigations for operator algebras and triples can be found in [1,2,[15][16][17]37].…”