Pedro J. Paúl

Barrelled subspaces of spaces with subseries decompositions or Boolean rings of projections

The research presented in this paper started by extending a theorem of Swetits [18]about barrelledness of subspaces of metrizable AK-spaces to general AK-spaces of scalar sequences. The extension reads as follows. (1) A subspace λ 0 of a barrelled AK-space λ such that λ 0 ⊃ φ is barrelled if and only if its dual is weak* sequentially complete. If in addition λ 0 is monotone, then it is barrelled if and only if equals the Kothe dual of λ 0 . As an easy consequence of this extension, we obtained the following result of Elstrodt and Roelcke [8, Corollary 3.4]. (2) If λ is a barrelled monotone AK-space, then also its subspace ℒ(λ), consisting of all sequences in λ with zero-density support, is barrelled .

An Abstract Banach-Steinhaus Theorem and Applications to Function Spaces

We give an abstract Banach-Steinhaus theorem for locally convex spaces having suitable algebras of linear projections modelled on a σ-finite measure space. This theorem is applied to deduce barrelledness results for the space L∞ (μ, E) of essentially bounded and μ-measurable functions from a Radon measure space (Ω, σ, μ) into a locally convex space E and also for B (μ, E), the closure of the space of simple functions. Sample: if μ is atomless, then B (μ, E) is barrelled if and only if E is quasi-barrelled and E′(β (E′, E)) has the property (B) of Pietsch.

Some new classes of Banach-Mackey spaces

Some weakenings of property (K) of Antosik for locally convex spaces are introduced: local property (K) and, for spaces with Schauder-type decompositions, two variants of property (K) defined in terms of block- and tail-sequences. It is shown that if a space enjoys any of these new properties, then it is Banach-Mackey. An application to the barrelledness of the spaces of Pettis integrable functions is given, and examples are provided to distinguish between the various K-type properties.

Properties of generalized Toeplitz operators

AbstractAn operatorX:

COMPLEMENTED COPIES OF c 0 IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

X:\mathcal{H}_1 \to \mathcal{H}_2

Barrelledness and bornological conditions on spaces of vector-valued μ-simple functions

is said to be a generalized Toeplitz operator with respect to given contractionsT1 andT2 ifX=T2XT1*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foiaş, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators. Namely, Wintners theorem of invertibility of analytic Toeplitz operators, Widom and Devinatzs invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintners theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintners estimate of the norm of a compactly perturbed Toeplitz operator, and the non-existence of compact classical Toeplitz operators due to Brown and Halmos.

Duals of vector-valued Köthe function spaces

For an order-continuous Banach function space Λ and a separated inductive limit E:= indnEn, we prove that indn A {En} is a topological subspace of Λ {E}; moreover, both spaces coincide if the inductive limit is hyperstrict. As a consequence, we deduce that if E is an LF-space, then Lp {E} is barrelled for 1 ≤ p ≤ ∞.

Quasi-similarity of contractions having a 2 × 1 characteristic function

Abstract Let X be a Banach space containing a copy of c0, then the space of Pettis integrable functions defined from any perfect atomless measure space to X, contains a complemented copy of c0.