Banach spaces of operators that are complemented in their biduals

Author： Delgado J.   Piñeiro C.  

Publisher： Springer Publishing Company

ISSN： 0236-5294

Source： Acta Mathematica Hungarica, Vol.115, Iss.1-2, 2007-04, pp. : 49-58

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Abstract

Let [A, a] be a normed operator ideal. We say that [A, a] is boundedly weak*-closed if the following property holds: for all Banach spaces X and Y, if T: X → Y** is an operator such that there exists a bounded net (T i ) i∈i in A(X, Y) satisfying lim i 〈Y*, T i X Y*〉 for every X ∈ X and Y* ∈ Y*, then T belongs to A(X, Y**). Our main result proves that, when [A, a] is a normed operator ideal with that property, A(X, Y) is complemented in its bidual if and only if there exists a continuous projection from Y** onto Y, regardless of the Banach space X. We also have proved that maximal normed operator ideals are boundedly weak*-closed but, in general, both concepts are different.

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