On the set of bounded linear operators transforming a certain sequence of a Hilbert space into an absolutely summable one

From the text: "Let H be a real, separable Hilbert space, B the set of bounded linear operators on H, and S={an:n∈N} a fixed sequence in H; we set CS={A∈B:∑∞n=1||Aan||<∞}. Obviously CS≠{0}, and it is easy to check that CS is a left ideal. Theorem 1: Let S={an:n∈N} be summable. Then CS contains a noncompletely continuous operator. Theorem 2: Let S={an:n∈N} be such that ∑∞n=1||an|||=∞; then there exists a completely continuous operator C not belonging to CS.'

Compact Hausdorff group topologies for the additive group of real numbers

This volume contains the contributions presented by several colleagues as a tribute to the mathematical and human qualities of Jos e Mar a Montesinos Amilibia on the occasion of his seventieth birthday. The editors would like to express their thanks to the contributors and their very especial gratitude to Jos e Mar a for his example through many years of scienti c and personal contactWe deal with an example of a topology for the additive group of real numbers R, which makes it a compact Hausdor topological group. Further, (R; ) is connected, but neither arcwise connected, nor locally connected. Thus, it is nei- ther a Lie group, nor a curve in the sense of H. Mazurkiewicz. The contribution of this short note is to provide an elementary proof of the fact that it is not arcwise connected.Peer ReviewedPostprint (published version

On strongly reflexive topological groups

Let Gˆ denote the Pontryagin dual of an abelian topological group G. Then G is reflexive if it is topologically isomorphic to Gˆˆ, strongly reflexive if every closed subgroup and every Hausdorff quotient of G and of Gˆ is reflexive. It is well known that locally compact abelian (LCA) groups are strongly reflexive. W. Banaszczyk [Colloq. Math. 59 (1990), no. 1, 53–57], extending an earlier result of R. Brown, P. J. Higgins and S. A. Morris [Math. Proc. Cambridge Philos. Soc. 78 (1975), 19–32], showed that all countable products and sums of LCA groups are strongly reflexive. L. Aussenhofer [Dissertationes Math. (Rozprawy Mat.) 384 (1999), 113 pp.] showed that all Čech-complete nuclear groups are strongly reflexive. It is an open question whether the strongly reflexive groups are exactly the Čech-complete nuclear groups and their duals. A Hausdorff topological group G is almost metrizable if and only if it has a compact subgroup K such that G/K is metrizable [W. Roelcke and S. Dierolf, Uniform structures on topological groups and their quotients, McGraw-Hill, New York, 1981]. In this paper it is shown that the annihilator of a closed subgroup of an almost metrizable group G is topologically isomorphic to the dual of the corresponding Hausdorff quotient, and an analogous statement holds for the character group of G. It then follows that an almost metrizable group is strongly reflexive only if its Hausdorff quotients and those of its dual are reflexive

Construcción de la topología de la convergencia débil en el espacio de Hilbert

Weak sequential convergence defines a topology by the requirement that a set O is open if xn⇀x and x∈O implies almost every xn∈O. This topology, Tc, is considered for a real Hilbert (separable) space. A form for the neighborhoods of the origin is given and Tc is compared with the weak topology. Tc is the finest topology that induces the same topology in bounded sets as the weak topology. Tc is an S-topology for S the set of pre-compact sets. The relation with other S-topologies is also give

Sobre algunos espacios de funciones continuas en el círculo unidad

This paper studies the topological group structure of C(X,T), the group of continuous functions on the topological space X with values in the circle group T, with the topology of uniform convergence on compact subsets of X. For the main part, attention is restricted to the case X=Q, the rational numbers with either the Euclidean or Bohr topologies. The style of the paper is largely expository, though some new results are proved. It is shown for instance that while the homomorphism group Hom(Q,T) (also known as Qˆ) is topologically isomorphic to Hom(R,T) (and, thus, to R), the group C(Q,T) is not even first countable. The group C(Q,T) is next realized as the completion of C(Qb,T), where Qb stands for the group Q equipped with its Bohr topology, the one induced by all continuous characters (homomorphisms into T) of Q. Another set of results concerns the duality properties of these groups. Here the authors represent C(Qb,T) as the character group of the free abelian topological group A(Qb,T) and exploit the duality properties of the latter to show that C(Qb,T) is a reflexive topological group

Group valued null sequences and metrizable non-Mackey groups

For a topological abelian group X we topologize the group c0(X) of all X-valued null sequences in a way such that when X= the topology of c0() coincides with the usual Banach space topology of the classical Banach space c0. If X is a non-trivial compact connected metrizable group, we prove that c0(X) is a non-compact Polish locally quasi-convex group with countable dual group c0(X). Surprisingly, for a compact metrizable X, countability of c0(X) leads to connectedness of X. Our principal application of the above results is to the class of locally quasi-convex Mackey groups (LQC-Mackey groups). A topological group (G,) from a class of topological abelian groups will be called a Mackey group in or a -Mackey group if it has the following property: if is a group topology in G such that (G,) and (G,) has the same character group as (G,), then . Based upon the results obtained for c0(X), we provide a large family of metrizable precompact (hence, locally quasi-convex) connected groups which are not LQC-Mackey. Namely, we show that for a connected compact metrizable group X0, the group c0(X), endowed with the topology induced from the product topology on X, is a metrizable precompact connected group which is not a Mackey group in LQC. Since metrizable locally convex spaces always carry the Mackey topology – a well-known fact from Functional Analysis –, our results prove that a Mackey theory for abelian groups is not a simple traslation of items known to hold for locally convex spaces. This paper is a contribution to the Mackey theory for groups, where properties of a topological nature like compactness or connectedness have an important role

Locally Quasi-Convex Compatible Topologies on a Topological Group

(ii) if (D) is a discrete abelian group of infinite rank, then (mathscr{C}(D)) is quasi-isomorphic to the poset (mathfrak{F}_D) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group (G ) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset ( mathscr{C} (G) ) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group (X), the group of null sequences (G=c_0(X)) with the topology of uniform convergence is studied. We prove that (mathscr{C}(G)) is quasi-isomorphic to (mathscr{P}(mathbb{R})) (6.9)