m at h . FA ] 3 1 Ja n 20 06 A HEREDITARILY INDECOMPOSABLE ASYMPTOTIC l 2 BANACH SPACE

Strictly Singular Non-compact Operators on Hereditarily Indecomposable Banach Spaces

An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic `1 Banach space. The construction of this operator relies on the existence of transfinite c0-spreading models in the dual of the space.

m at h . FA ] 3 0 Ja n 20 06 Pointwise Values and Fundamental Theo - rem in the Algebra of Asymptotic Func - tions

ar X iv : m at h / 06 06 55 1 v 1 [ m at h . FA ] 2 2 Ju n 20 06 COMPLEX INTERPOLATION OF COMPACT OPERATORS

If (A 0 , A 1) and (B 0 , B 1) are Banach couples and a linear operator T : A 0 + A 1 → B 0 + B 1 maps A 0 compactly into B 0 and maps A 1 boundedly into B 1 , does T necessarily also map [A 0 , A 1 ] θ compactly into [B 0 , B 1 ] θ for θ ∈ (0, 1)? After 42 years this question is still not answered, not even in the case where T : A 1 → B 1 is also compact. But affirmative answers are known for ...

Interpolating Hereditarily Indecomposable Banach Spaces

A Banach space X is said to be Hereditarily Indecomposable (H.I.) if for any pair of closed subspaces Y , Z of X with Y ∩ Z = {0}, Y + Z is not a closed subspace. (Throughout this section by the term “subspace” we mean a closed infinite-dimensional subspace of X .) The H.I. spaces form a new and, as we believe, fundamental class of Banach spaces. The celebrated example of a Banach space with no...

ar X iv : m at h / 05 06 21 5 v 1 [ m at h . FA ] 1 1 Ju n 20 05 Scaled Enflo type is equivalent to Rademacher type Manor Mendel

We introduce the notion of scaled Enflo type of a metric space, and show that for Banach spaces, scaled Enflo type p is equivalent to Rademacher type p.