A weak Hilbert space with few symmetries

Abstract

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.

Operators in tight by support Banach spaces

“…Namely we prove the following version of Prop. 7.5 and 7.6 of [2] in case of Gowers unconditional space, generalizing Theorem 29 [12]. Proposition 2.11.…”

“…It is known that any bounded operator on the whole space X U is a strictly singular perturbation of a diagonal operator [11]. Adapting arguments of [2] one can prove analogous result for any bounded operator T : Y → Y , where Y is a block subspace of X U . W.T.…”

“…[(cf. Lemma 7.2 [2])] For any partitions A n = B n ∪ C n , n ∈ N, we have lim n R Cn T R Bn x n = 0.…”

Banach spaces without minimal subspaces – Examples

“…In this section we prove that the dual of the type (3) space G u constructed by Gowers in [12] is locally minimal of type (3), that Gowers' hereditarily indecomposable and asymptotically unconditional space G defined in [13] is of type (1), and that its dual G * is locally minimal of type (1). These spaces are natural variations on Gowers and Maurey's space GM , and so familiarity with that construction will be assumed: we shall not redefine the now classical notation relative to GM , such as the sets of integers K and L, rapidly increasing sequences (or R.I.S.…”

On Uniformly Finitely Extensible Banach spaces

“…This answers the question left open in [18, p.2131] of whether weak Hilbert spaces must be UFO. Argyros, Beanland and Raikoftsalis [4] have recently constructed a weak Hilbert space X abr with an unconditional basis in which no disjointly supported subspaces are isomorphic (such spaces are called tight by support in [24]). Clearly this space does not contain a copy of ℓ 2 .…”