Abstract. We introduce and study the Bourgain index of an operator between two Banach spaces. In particular, we study the Bourgain ℓp and c0 indices of an operator. Several estimates for finite and infinite direct sums are established. We define classes determined by these indices and show that some of these classes form operator ideals. We characterize the ordinals which occur as the index of an operator and establish exactly when the defined classes are closed. We study associated indices for non-preservation of ℓ ξ p and c ξ 0 spreading models and indices characterizing weak compactness of operators between separable Banach spaces. We also show that some of these classes are operator ideals and discuss closedness and distinctness of these classes.
Abstract. Let X and Y be separable Banach spaces. Suppose Y either has a shrinking basis or Y is isomorphic to C(2 N ) and A is a subset of weakly compact operators from X to Y which is analytic in the strong operator topology. We prove that there is a reflexive space with a basis Z such that every T ∈ A factors through Z. Likewise, we prove that if A ⊂ L(X, C(2 N )) is a set of operators whose adjoints have separable range and is analytic in the strong operator topology then there is a Banach space Z with separable dual such that every T ∈ A factors through Z. Finally we prove a uniformly version of this result in which we allow the domain and range spaces to vary.
We prove quantitative factorization results for several classes of operators, including weakly compact, Rosenthal, and ξ-Banach-Saks operators.
Abstract.A hereditarily indecomposable asymptotic 2 Banach space is constructed. The existence of such a space answers a question of B. Maurey and verifies a conjecture of W. T. Gowers.2000 Mathematics Subject Classification. 46B20, 46B03. Introduction.A famous open problem in functional analysis is whether there exists a Banach space X such that every (bounded linear) operator on X has the form λ + K where λ is a scalar and K denotes a compact operator. This problem is usually called the "scalar-plus-compact" problem [14]. One of the reasons this problem has become so attractive is that by a result of N. Aronszajn and K. T. Smith [7], if a Banach space X is a solution to the scalar-plus-compact problem then every operator on X has a non-trivial invariant subspace and hence X provides a solution to the famous invariant subspace problem. An important advancement in the construction of spaces with "few" operators was made by W. T. Gowers and B. Maurey [16], [17]. The ground breaking work [16] provides a construction of a space without any unconditional basic sequence thus solving, in the negative, the long standing unconditional basic sequence problem. The Banach space constructed in [16] is Hereditarily Indecomposable (HI), which means that no (closed) infinite dimensional subspace can be decomposed into a direct sum of two further infinite dimensional subspaces. It is proved in [16] that if X is a complex HI space then every operator on X can be written as λ + S where λ is a scalar and S is strictly singular (i.e. the restriction of S on any infinite dimensional subspace of X is not an isomorphism). It is also shown in [16] that the same property remains true for the real HI space constructed in [16]. V. Ferenczi [10] proved that if X is a complex HI space and Y is an infinite dimensional subspace of X then every operator from Y to X can be written as λi Y + S where i Y : Y → X is the inclusion map and S is strictly singular. It was proved in [17] that, roughly speaking, given an algebra of operators satisfying certain conditions, there exists a Banach space X such that for every infinite dimensional subspace Y , every operator from Y to X can be written as a strictly singular perturbation of a restriction to Y of some element of the algebra.The construction of the first HI space prompted researchers to construct HI spaces having additional nice properties. In other words people tried to "marry" the exotic structure of the HI spaces to the nice structure of classical Banach spaces.
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