Abstract. Motivated by a problem of P. Koszmider we introduce the class of quasi-prime Banach spaces. This class lies between the classes of prime and primary Banach spaces. It is shown that for every 1 < p < ∞ there exists a strictly quasi-prime separable reflexive Banach space X p such that p is a complemented subspace of X p . A similar result also holds for the case of 1 and c 0 . More generally, for every separable decomposable prime Banach space Y not containing 1 there exists a strictly quasi-prime X Y containing Y as a complemented subspace. We also investigate the operators acting on these spaces as well as the complemented subspaces of their finite powers.
We construct a separable Banach space X wh with an unconditional basis that is a weak Hilbert space and no block subspace is linearly isomorphic to any of its proper subspaces.We prove that the space X wh satisfies these properties by showing it is strongly asymptotic 2 and that every bounded linear operator on X wh is a strictly singular perturbation of a diagonal operator with respect to the unit vector basis.
r é s u m éNous construisons un space de Banach X wh qui est un espace failble de Hilbert et n'admettant aucune sous-espace bloc isomorphe linéaire à une sous-espace. Nous démontrons les propriétés de X wh par démontrons que X wh est fortement asymptotique 2 et tout opérateur borné de X wh soit une variation strictment singulière d'un opérateur diagonal par rappert à la base.
Abstract. We construct a weak Hilbert Banach space such that for every block subspace Y every bounded linear operator on Y is of the form D + S, where S is a strictly singular operator and D is a diagonal operator. We show that this yields a weak Hilbert space whose block subspaces are not isomorphic to any of their proper subspaces.
Abstract. It is shown that every separable reflexive Banach space is a quotient of a reflexive Hereditarily Indecomposable space, which yields that every separable reflexive Banach is isomorphic to a subspace of a reflexive Indecomposable space. Furthermore, every separable reflexive Banach space is a quotient of a reflexive complementably ℓp saturated space with 1 < p < ∞ and of a c 0 saturated space.
Abstract. The famous Gowers tree space is the first example of a space not containing c0, 1 or a reflexive subspace. We present a space with a similar construction and prove that it is hereditarily indecomposable (HI) and has 2 as a quotient space. Furthermore, we show that every bounded linear operator on it is of the form λI + W where W is a weakly compact (hence strictly singular) operator.
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