Vector lattice covers of ideals and bands in Pre-Riesz spaces

Abstract

Pre-Riesz spaces are ordered vector spaces which can be order densely embedded into vector lattices, their so-called vector lattice covers. Given a vector lattice cover Y for a pre-Riesz space X, we address the question how to find vector lattice covers for subspaces of X, such as ideals and bands. We provide conditions such that for a directed ideal I in X its smallest extension ideal in Y is a vector lattice cover. We show a criterion for bands in X and their extension bands in Y as well. Moreover, we state properties of ideals and bands in X which are generated by sets, and of their extensions in Y.     

The functional model of a j-unitary node with a given j-inner characteristic matrix function

The known model of aj-unitary node with a givenj-inner characteristic matrix functionW is obtained from the known model of a unitary node with a given bi-inner characteristic matrix functionS, using the Potapov-Ginzburg transforms of the nodes and their characteristic functionsW andS. We show that some new properties of the L. de Branges reproducing kernal Hilbert spaces (W) which were discovered by H. Dym are characteristic properties of these spaces.     

On metrizable non-Archimedean LF-spaces

It is known that no non-Archimedean LB-space (and no strict non-Archimedean LF-space) is metrizable. We show that there exist many metrizable (or even normable) non-Archimedean LF-spaces. We prove that every non-normable polar non-Archimedean Fréchet space (and every non-Archimedean Banach space with an infinite basis (x_α)) contains a dense subspace which is an LF-space.     

Sequential convergence in topological vector spaces

For a given linear topology , on a vector spaceE, the finest linear topology having the same convergent sequences, and the finest linear topology onE having the same precompact sets, are investigated. Also, the sequentially bornological spaces and the sequentially barreled spaces are introduced and some of their properties are studied.     

Some Results on Best Coapproximation in L 1 (S, X)

As a counterpart to best approximation in normed linear spaces, best coapproximation was introduced by Franchetti and Furi. In this paper, we shall show that if M is a separable, coproximinal subspace of X satisfying some additional conditions, then L ¹ (S, M) is coproximinal in L ¹(S, X).      

Disjointness in Partially Ordered Vector Spaces

A notion of disjointness in arbitrary partially ordered vector spaces is introduced by calling two elements x and y disjoint if the set of all upper bounds of x + y and −x − y equals the set of all upper bounds of x − y and −x + y. Several elementary properties are easily observed. The question whether the disjoint complement of a subset is a linear subspace appears to be more difficult. It is shown that in directed Archimedean spaces disjoint complements are always subspaces. The proof relies on theory on order dense embedding in vector lattices. In a non-Archimedean directed space even the disjoint complement of a singleton may fail to be a subspace. According notions of disjointness preserving operator, band, and band preserving operator are defined and some of their basic properties are studied.     

Reproducing kernels for harmonic Bergman spaces of the unit ball

We study reproducing kernels for harmonic Bergman spaces of the unit ball inR ⁿ . We establish some new properties for the reproducing kernels and give some applications of these properties.     

The criteria of strict monotonicity and rotundity points in generalized Calderón–Lozanovskiǐ spaces

In this paper, some basic properties of the general modular space are proven. Criteria for strictly monotone points, extreme points and SU$SU$-points in generalized Calderón–Lozanovskiǐ spaces are obtained. Consequently, the sufficient and necessary conditions for the rotundity properties of such spaces are given.     

MULTIPLIERS OF MATRIX SPACES

Abstract

Suppose X and Y are FK spaces in which ? the span of the coordinate vectors (e_n) is dense. Let L(X,Y) denote the space of all matrices of the form E_i(T(e_j)) as T ranges over all continuous linear operators from X into Y; here e_i represents the i^th coordinate vector and E_i represents the i^th coordinate functional. Let M(L(X, Y)) denote the space of all matrices B such that (B(i,j)A(i,j)) is in L(X,Y) whenever A is in L(X,Y). In this paper we shall show how the summability properties of X and Y determine the extent of M(L(X,Y)) and conversely how the extent of M(L(X,Y)) determines the summability properties of both X and Y.     

ON VECTOR SPACES WITH LOCALLY BOUNDED CONVERGENCE STRUCTURES

ABSTRACT

It is known that a precompact set in a limit vector space is not necessarily bounded. In this paper it is shown that the topological vector space result that every precompact set is bounded can be extended to the large class of locally bounded prelimit vector spaces. This result is used to extend the well-known characterization of finite-dimensional separated topological vector spaces to locally bounded prelimit vector spaces.     

Applications of a decomposition inX⊗λ Y

A Schauder Decomposition inX⊗_λ when Y has a Schauder Decomposition is presented. This decomposition is then used to investigate the Approximation Property. For instance, it is shown that if X has the Approximation Property and Y has a basis thenX⊗_λ has the Approximation Property. We also use this to obtain results concerning operators on vector valued continuous function spaces. Entrata in Redazione il 31 marzo 1999     

A conglomerate of exponential supercategories of the category of finitely generated topological spaces

Summary For every ordinaln > 1 we define a categoryT _n of topological spaces in ech's sense which is isomorphic to a category ofn-ary monorelational systems. We show that every categoryT _n is an exponential supercategory of the categoryB of finitely generated topological spaces, which means that well-behaved function spacesG ^H can be defined inT _n wheneverG B.     

Banach-stone theorems and separating maps

LetC(X,E) andC(Y,F) denote the spaces of continuous functions on the Tihonov spacesX andY, taking values in the Banach spacesE andF, respectively. A linear mapH:C(X,E)→C(Y,F) isseparating iff(x)g(x)=0 for allx inX impliesHf(y)Hg(y)=0 for ally inY. Some automatic continuity properties and Banach-Stone type theorems (i.e., asserting that isometries must be of a certain form) for separating mapsH between spaces of real- and complex-valued functions have already been developed. The extension of such results to spaces of vector-valued functions is the general subject of this paper. We prove in Theorem 4.1, for example, for compactX andY, that a linear isometryH betweenC(X,E) andC(Y,F) is a “Banach-Stone” map if and only ifH is “biseparating (i.e,H andH ⁻¹ are separating). The Banach-Stone theorems of Jerison and Lau for vector-valued functions are then deduced in Corollaries 4.3 and 4.4 for the cases whenE andF or their topological duals, respectively, are strictly convex. Research supported by the Fundació Caixa Castelló, MI/25.043/92     

The Canonical Spectral Measure in Köthe Echelon Spaces

Operator and measure theoretic properties of the canonical spectral measure acting in K?the echelon sequence spaces X are characterized via topological and geometric properties of X (such as being nuclear, Montel, satisfying the density condition, etc.).     

Categories of abstract smooth models and their singular envelopes

We define the categories of (abstract) smooth models (Definition 1.2) and, in the additive case, their singular envelopes (Definition 1.5). The first main result is a relative version of the Yoneda representation theorem (Theorem 1.6), and the second one is an existence and uniqueness theorem for the singular envelope (Theorem 1.7). In fact we prove the existence of a canonical process which associates with each additive smooth-model categoryS a singular envelopeS-an ofS, whose objects are calledS-analytic spaces (Definition 5.1). We notice that most of the fundamental categories of geometry are of the formS-an (up to equivalence). As an application we introduce here two such categories: Banach differentiable spaces and Banach mixed spaces.The author is indebted to the Department of Mathematics of the University of Rome (La Sapienza) for hospitality and financial support and to the referees for their critical comments on the first version of this paper.     

Smooth points in some spaces of bounded operators

We determine the smooth points of certain spaces of bounded operatorsL(X,Y), including the cases whereX andY arel ^p -orc ₀-direct sums of finite dimensional Banach spaces or subspaces of the latter enjoying the metric compact approximation property. We also remark that the operators not attaining their norm are nowhere dense inL(X,Y) wheneverK(X,Y) is anM-ideal inL(X,Y).     

Wick calculus in Gaussian analysis

We define an extension of the distribution spaces conventionally used in Gaussian analysis. This space, characterized by analytic properties of S-transforms, allows for a calculus based on the Wick product. We indicate some of its features.     

Mackey Topologies and Mixed Topologies in Riesz Spaces

The possibility of characterizing the Mackey topology of a dual pair of vector spaces as a generalized inductive limit (or mixed) topology is investigated. Positive answers are given for a wide range of dual pairs of Riesz spaces (vector lattices) and non-commutative Banach function spaces (or symmetric operator spaces).