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.     

Topologically Algebraic Algebras

We show that a locally convex algebra is topologically algebraic if, and only if, it is algebraic.Thanks are offered to Professor M. Oudadess for many remarks which allowed an improvement of the first version of this paper. We are also indebted to Professor A. Mallios for valuable suggestions.     

Carleson Measures for Spaces of Dirichlet Type

If 0 < p < ∞ and α > − 1, the space consists of those functions f which are analytic in the unit disc and have the property that f ′ belongs to the weighted Bergman space A_α^p. In 1999, Z. Wu obtained a characterization of the Carleson measures for the spaces for certain values of p and α. In particular, he proved that, for 0 < p ≤ 2, the Carleson measures for the space are precisely the classical Carleson measures. Wu also conjectured that this result remains true for 2 < p < ∞. In this paper we prove that this conjecture is false. Indeed, we prove that if 2 < p < ∞, then there exists g analytic in such that the measure μ_g,p on defined by dμ_g,p (z) = (1 − |z|²)^{p - 1}| g ′ (z)|^p dx dy is not a Carleson measure for but is a classical Carleson measure. We obtain also some sufficient conditions for multipliers of the spaces      

Representation of a class of locally convex (M)-lattices

We prove a representation theorem for Hausdorff locally convex (M)-lattices which are Dedekind σ-complete, and whose topologies are order σ-continuous and monotonically complete. These turn out to be the weighted spaces c₀(T, H), defined in the paper for T ≠ and H ^T₊. We also characterize the dual of c₀(T, H), as the space l¹ (T, H) defined in the last section. The known representation (on c₀(T)) of Banach (M)-lattices with order continuous norm follows as a particular case. We obtain these results by first proving a new general isomorphism theorem, which seems to be of independent interest. Our notion of “monotonic topological completeness” is weaker than the usual completeness and seems to be very convenient in the framework of topological ordered vector spaces.     

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.     

Spectral measures,II: Characterization of scalar operators

We continue the development of part I. The Riesz representation theorem is proved without assuming local convexity. This theorem is applied to give sufficient conditions for an operator (continuous or otherwise) to be spectral. A uniqueness problem is pointed out and the function calculus is extended to the case of several variables. A Radon—Nikodym theorem is proved.     

New double sequence spaces of fuzzy numbers

Abstract

In this paper we introduce some new double sequence spaces of fuzzy numbers using Orlicz function and also we examine that if a sequence of fuzzy numbers is double strongly convergent with respect to an Orlicz function then it is double statistical convergent.     

Spectral measures,III: Densely defined spectral measures

In this paper we extend the theory of spectral measures developed in Parts I and II to the case where values are assumed in the set of discontinuous (in normed spaces „unbounded”) operators. Examples of operators in nonlocally convex spaces are given, which have densely defined measures.     

Wick Product and Backward Heat Equation

In the present paper the Wick version of analytic functions with respect to a one dimensional Brownian motion is shown to be closely related to the backward heat equation. This fact provides representation theorems for a certain class of random variables in terms of Wick powers. In addition, we obtain explicit formulas for the action of some second quantization operators arising in the applications.     

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.     

Polarizations of Prym varieties of pairs of coverings

To any pair of coverings f_i:X → X_i, i=1, 2, of smooth projective curves one can associate an abelian subvariety of the Jacobian J_X, the Prym variety P(f₁, f₂) of the pair (f₁, f₂). In some cases we can compute the type of the restriction of the canonical principal polarization of J_X. We obtain 2 families of Prym-Tyurin varieties of exponent 6. Received: 2 September 2004     

The Structure of the Closure of the Rational Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>(μ)

Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K°. Let μ be a positive finite measure with its support contained in K. For 1 ≤ q < ∞, let A^q(K, μ) denote the closure of A(K) in L^q(μ). The aim of this work is to study the structure of the space A^q(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for A^q(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for P^q(μ), the closure of polynomials in L^q(μ), as the special case when K is a closed disk containing the support of μ.     

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 Galois-theoretic approach to Kanev’s correspondence

Let G be a finite group, an absolutely irreducible -module and w a weight of . To any Galois covering with group G we associate two correspondences, the Schur and the Kanev correspondence. We work out their relation and compute their invariants. Using this, we give some new examples of Prym–Tyurin varieties. This work was supported by FONDECYT No. 11060468 and No. 1060742.     

On the Essential Lower Bound of Elements in von Neumann Algebras

In this paper we obtain a number of interesting results on the connection that exists between the essential lower bound of an element in a von Neumann algebra and its Fredholm properties. These connections lead to an extension of a well-known perturbation theorem on Fredholm operators of index zero. Finally we obtain for special cases necessary and sufficient conditions for elements in the closure of the group of invertible elements to be essentially topological divisor of zero     

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.     

The Krein Spectral Shift Function in Semifinite von Neumann Algebras

We show the existence of a spectral shift function in the sense of Krein for bounded trace class perturbations of a self-adjoint operator affiliated with a semifinite von Neumann algebra