Browsing by Author "Musundi, S."

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    Equivalent banach operator ideal norms
    (International journal of mathematical analysis, 2012) Musundi, S.; Aywa, Shem.; Fourie, J.
    We continue the investigation of coorbit spaces which can be attached to every integrable, irreducible, unitary representation of a locally compact groupG and every reasonable function space onG. Whereas Part I was devoted to atomic decompositions of such spaces, Part II deals with general properties of these spaces as Banach spaces. Among other things we show that inclusions, the quality of embeddings, reflexivity and minimality and maximality of coorbit spaces can be completely characterized by the same properties of the corresponding sequence spaces. In concrete examples (cf. Part III) one recovers several and often difficult theorems with ease.
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    Unconditional banach space ideal property
    (Journal of mathematical sciences advances and applications, 2012) Musundi, S.; Aywa, Shem.; Fourie, Jan.; Matuya, W.J.; Godefroy, G.
    A Banach X is an M-ideal in its bidual if the relation \\y^t\\ = \\y\\ + 11^1 1 holds for every y € X* and every t e X1 £ j^***. The spaces Co(J) — / any set-equipped with their canonical norm belong to this class, which also contains e.g. certain spaces K(E,F) of compact operators between reflexive spaces (see [11]) and certain spaces of the form C(G)/CA (G) where G is an abelian compact group and A is a subset of the discrete dual group (see [5]). This class has been carefully investigated, in particular by A. Lima and by the «West-Berlin school», since the notion of Af-ideal was introduced by Alfsen and Effros in 1972 [1]. We will show in this paper that these spaces somehow «look like » C o; more precisely, that they share the property (u) with this latter space. This solves affirmatively a question that was pending for several years, and provides improvements of some results of [6] and [10]. Our proof uses non-linear arguments. The key lemma is actually a special case of a fundamental lemma ([I], lemma 1.4.) of the original article of Alfsen and Effros. Notation. — The closed unit ball of a Banach space Z is denoted by Zi, and its dual by Z*. The topology defined on Z* by the pointwise convergence on Z is denoted by co*. The canonical injection from a Banach space X into its bidual X** is denoted by i. A sequence (Xm) in X is said to be a weakly unconditionally convergent series —