We introduce an ordinal index which measures the complexity of a weakly null sequence, and show that a construction due to J. Schreier can be iterated to produce for each α < ω₁, a weakly null sequence in with complexity α. As in the Schreier example each of these is a sequence of indicator functions which is a suppression-1 unconditional basic sequence. These sequences are used to construct Tsirelson-like spaces of large index. We also show that this new ordinal index is related to the Lavrent’ev...
We construct an indecomposable reflexive Banach space such that every infinite-dimensional closed subspace contains an unconditional basic sequence. We also show that every operator is of the form λI + S with S a strictly singular operator.
We present a characterization of continuous surjections, between compact metric spaces, admitting a regular averaging operator. Among its consequences, concrete continuous surjections from the Cantor set 𝓒 to [0,1] admitting regular averaging operators are exhibited. Moreover we show that the set of this type of continuous surjections from 𝓒 to [0,1] is dense in the supremum norm in the set of all continuous surjections. The non-metrizable case is also investigated. As a consequence, we obtain...
It is shown that for every k ∈ ℕ and every spreading sequence eₙₙ that generates a uniformly convex Banach space E, there exists a uniformly convex Banach space admitting eₙₙ as a k+1-iterated spreading model, but not as a k-iterated one.
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 -saturated space with and of a saturated space.
We present an example of a Banach space admitting an equivalent weakly uniformly rotund norm and such that there is no , for any set , linear, one-to-one and bounded. This answers a problem posed by Fabian, Godefroy, Hájek and Zizler. The space is actually the dual space of a space which is a subspace of a WCG space.
2000 Mathematics Subject Classification: Primary 46E15, 54C55; Secondary 28B20.
For weakly compact subsets of Hilbert spaces K, we study the
existence of totally disconnected spaces L, such that C(K) is isomorphic
to C(L).
We prove that the space C(BH ) admits a Pełczyński decomposition and
we provide a starshaped weakly compact K, subset of BH with non-empty
interior in the norm topology, and such that C(K) ~= C(L) with L totally disconnected.
Research partially supported...
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