WEAK and NORM COMPACTNESS ON KÖTHE-BOCHNER FUNCTION SPACES

Abstract

We present some characterizations of weak, weak conditional and norm compactness in spaces of vector valued order continuous Köthe function spaces. These characterizations use oscillation conditions over functions and regular methods of summability.     

On the arc component of a locally compact Abelian group

For a topological group G, we denote by G _a the arc component of the neutral element and by the character group of G, i.e. the group of all continuous homomorphisms from G into T. We prove the following theorem: Let G be a connected locally compact abelian group and let be the embedding. Then is a topological isomorphism. In particular, the character group of the arc component of a compact abelian group is discrete. Some conclusions will be drawn.     

An isomorphic characterization of L-spaces

We show that a sequentially (τ)-complete topological vector lattice X_τ is isomorphic to some L¹(μ), if and only if the positive cone can be written as X₊ = ₊B for some convex, (τ)-bounded, and (τ)-closed set B X₊ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y₊ = ₊ (for u >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.     

On stability of the Cauchy equation on semigroups

Summary We say that Hyers's theorem holds for the class of all complex-valued functions defined on a semigroup (S, +) (not necessarily commutative) if for anyf:S such that the set {f(x + y) – f(x) – f(y): x, y S} is bounded, there exists an additive functiona:S for which the functionf – a is bounded.Recently L. Székelyhidi (C. R. Math. Rep. Acad. Sci. Canada8 (1986) has proved that the validity of Hyers's theorem for the class of complex-valued functions onS implies its validity for functions mappingS into a semi-reflexive locally convex linear topological spaceX. We improve this result by assuming sequential completeness of the spaceX instead of its semi-reflexiveness. Our assumption onX is essentially weaker than that of Székelyhidi. Theorem.Suppose that Hyers's theorem holds for the class of all complex-valued functions on a semigroup (S, +) and let X be a sequentially complete locally convex linear topological (Hausdorff) space. If F: S X is a function for which the mapping (x, y) F(x + y) – F(x) – F(y) is bounded, then there exists an additive function A : S X such that F — A is bounded.     

Unconditional bases and the order structure

The paper deals with an analytic characterization of unconditional bases, a characterization of the unconditional character of a Schauder base for the associated cone and a number of counterexamples supporting various conditions appearing in the theorems.     

A basic strict separation theorem in random locally convex modules

The central idea of this paper is to make full use of the recently developed theory of random conjugate spaces to establish a basic strict separation theorem that is universally suitable in an arbitrary random locally convex module. A series of interesting corollaries of the basic theorem are also included.     

On products of additive functions (a third approach)

Summary Leta ₁, , a_s : G K be additive functions from an abelian groupG into a fieldK such thata ₁(g)··a_s(g) = 0 for allg G. If char(K) =0, then it is well known that one of the functions a₁ has to vanish. We give a new proof of this result and show that, if char(K) > 0, it is only valid under additional assumptions.     

Results of Vidav-Palmer Type in m-Convex Algebras

Using the notion of a normal cone, we give results of Vidav-Palmer type in terms of hermitian, quasi-hermitian and uniformly quasi-hermitian elements. Consequences are also given.     

Spectral measures,I: General theory in a topological vector space

Part I of this paper is devoted to the general theory of spectral measures in topological vector spaces. We extend the Hilbert space theory to this setting and generalize the notion of spectral measure in some useful ways to provide a framework for Part II, etc.     

Rearrangement Invariant Continuous Linear Functionals on Weak <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>

We show that in the dual of Weak L¹ the subspace of all rearrangement invariant continuous linear functionals is lattice isometric to a space L¹(μ) and is the linear hull of the maximal elements of the dual unit ball. We also show that the dual of Weak L¹ contains a norm closed weak* dense ideal which is lattice isometric to an ℓ¹-sum of spaces of type C(K). Helmut H. Schaefer in memoriam     

A remark on the representation of vector lattices as spaces of continuous real-valued functions

The well-known Ogasawara-Maeda-Vulikh representation theorem asserts that for each Archimedean vector lattice L there exists an extremally disconnected compact Hausdorff space , unique up to a homeomorphism, such that L can be represented isomorphically as an order dense vector sublattice of the universally complete vector lattice C () of all extended-real-valued continuous functions f on for which % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaacmqabaGaeqyYdCNaeyicI4SaeyyQdCLaeyOoaOJaaiiFaiab% gkzaMkabgIcaOiabgM8a3jabgMcaPiaacYhacqGH9aqpcqGHEisPai% aawUhacaGL9baaaaa!4E05!\[\left\{ {\omega \in \Omega :|f(\omega )| = \infty } \right\}\] is nowhere dense. Since the early days of using this representation it has been important to find conditions on L such that consists of bounded functions only.The aim of this short article is to present a simple complete characterization of such vector lattices.     

On Two Variable Jordan Block (II)

On the Hardy space over the bidisk H²(D²), the Toeplitz operators and are unilateral shifts of infinite multiplicity. A closed subspace M is called a submodule if it is invariant for both and . The two variable Jordan block (S₁, S₂) is the compression of the pair to the quotient H²(D²) ⊖M. This paper defines and studies its defect operators. A number of examples are given, and the Hilbert-Schmidtness is proved with good generality. Applications include an extension of a Douglas-Foias uniqueness theorem to general domains, and a study of the essential Taylor spectrum of the pair (S₁, S₂). The paper also estabishes a clean numerical estimate for the commutator [S₁*, S₂] by some spectral data of S₁ or S₂. The newly-discovered core operator plays a key role in this study.     

Automatic continuity of n-homomorphisms between FrÉchet algebras

We impose a condition on a commutative regular Fréchet algebra (A, (p_m )) to ensure that A/kerp_m is a Fréchet Q-algebra. This implies that if θ is an n-homomorphism on certain Fréchet algebras (A, (p_m )) into semisimple commutative Fréchet algebras (B,(q_m)) such that θ(kerp_m) ? kerq_m, for large enough m, then θ is continuous. We also show that if A is a Fréchet Q-algebra, B is a semisimple Fréchet algebra, θ: A → B is a dense range n-homomorphism such that θ(A) is factorizable, and the spectral radius v_B is continuous on the separating space (θ), then θ is automatically continuous.     

A normalization problem for a class of singular integral operators with carleman shift and unbounded coefficients

The solution to a normalization problem for singular integral operators with Carleman shift and degenerate and unbounded coefficients inL _p () is obtained, where is either the unit circle or the real line. The approach followed consists mainly in two steps: the reduction to a singular integral operator with bounded coefficients and the use of the solution to an abstract normalization problem.This research was supported by JNICT under the grant PBIC/C/CEN/1040/92.