Abstract. A Banach space X is called an HT space if the Hilbert transform is bounded from L(X) into L(X), where 1 < p < ∞. We introduce the notion of an ACF Banach space, that is, a Banach space X for which we have an abstract M. Riesz Theorem for conjugate functions in L(X), 1 < p < ∞. Berkson, Gillespie, and Muhly [5] showed that X ∈ HT =⇒ X ∈ ACF. In this note, we will show that X ∈ ACF =⇒ X...

We study the range transformations 0p(/fD, Reß) and Op(/fD, B) for Banach function algebras A and B. As a special instance, the harmonicity of functions in Op(/fD, Re A) for a nontrivial function algebra A is established and is compared with previous investigations of Op( An, A) and Op((Re/l);, (Re/1)) for an interval /. In §2 we present some results on Op(AD, B) and use them to show that funct...

The complemented subspace problem asks, in general, which closed subspaces M of a Banach space X are complemented; i.e. there exists a closed subspace N of X such that X = M ⊕ N? This problem is in the heart of the theory of Banach spaces and plays a key role in the development of the Banach space theory. Our aim is to investigate some new results on complemented subspaces, to present a history...

The two main results are: A. If a Banach space X is complementably universal for all subspaces of c0 which have the bounded approximation property, then X∗ is non separable (and hence X does not embed into c0), B. There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X. Theorem B solves a problem that dates from the 1970s.

A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the π-property in terms of norms of compositions of projections is found. 2000 Mathematics Subject Classification. Primary 46B15; Secondary 46B07, 46B28. The problem of existence of Banach spaces with the π-property but without a finite dimensional decomposition is one of the...

This paper presents a rigidity theorem for infinite dimensional Bergman spaces of hyperbolic Riemann surfaces, which states that the Bergman space A1(M), for such a Riemann surface M , is isomorphic to the Banach space of summable sequence, l1. This implies that whenever M and N are Riemann surfaces which are not analytically finite, and in particular are not necessarily homeomorphic, then A1(M...

We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following: let X be a Banach space with an unconditional basis {ei}i∈N; then either there exists a perfect set P of infinite subsets of N such that for any two distinct A,B ∈ P, [ei]i∈A ≇ [ei]i∈B , or for a residual set of infinite subsets A of N, [ei]i∈A is isomorphic to X, and in that case, X is is...

We extend Troitsky’s ideas on measure-free martingales on Banach lattices to martingales of operators acting between a Banach lattice and a Banach space. We prove that each norm bounded martingale of cone absolutely summing (c.a.s.) operators (also known as 1-concave operators), from a Banach lattice E to a Banach space Y , can be generated by a single c.a.s. operator. As a consequence, we obta...