The Szlenk index and local ?1-indices

Abstract

Abstract. We introduce two new local ℓ1-indices of the same type as the Bourgain ℓ1-index; the ℓ + 1 -index and the ℓ + 1 -weakly null index. We show that the ℓ + 1 -weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain ℓ1. The ℓ + 1 -weakly null index has the same form as the Bourgain ℓ1-index: if it is countable it must take values ω α for some α < ω1.The different ℓ1-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its ℓ1-index, via the ℓ + 1 -weakly null index. We show that I(C(ω ω α )) = ω 1+α+1 .

Operators on $C(K)$ spaces preserving copies of Schreier spaces

“…We shall also sketch a proof for the calculation of the Szlenk index of the Schreier spaces X ξ in Corollary 3.4 (cf. also [10]). To achieve these goals, we shall use trees in order to find suitable representations for elements in the ξ-th Szlenk set of a w * -compact subset of B X * (X a separable Banach space).…”

“…If we take F = S ξ , the generalized ξ-th Schreier family, ξ < ω 1 , [6], [11], then X F is the Schreier space X ξ of order ξ. It is known that the Szlenk index of X ξ is ω ξ+1 (the proof of this fact is implicit in the arguments of [10]; see also Corollary 3.4) and thus equals the Szlenk index of C(ω ω ξ ). Therefore, in terms of the Szlenk index, X ξ is comparable to C(ω ω ξ ) and has an unconditional basis.…”

The universality of ℓ 1 as a dual space

“…If S z (X ) < ω 1 then S z (X ) = ω β for some β < ω 1 . Much has been written on the Szlenk index (e.g., see [3,6,[12][13][14]20,21,26]). We note that the upper and lower estimates in both theorems are with respect to the unit vector basis (t i ) of T c,α and its biorthogonal sequence (t * i ), a basis for T * c,α .…”

Weakly null sequences in the Banach space C(K)

“…Clearly, G α is a clopen subset of F for every α ∈ F. The sequence (χ Gα n ) ∞ n=1 is called the node basis of C(F). It is not hard to check that (χ Gα n ) ∞ n=1 is a normalized, monotone, shrinking Schauder basis for C(F) [3]. Proposition 3.1.…”

“…Hence, our assumption that N 0 contained no infinite subset which is α-nice, was false. The proof of the theorem will be completed, once we give the construction of the above described families, satisfying conditions (1)- (3). N 1 and β 1 have been already constructed.…”