Best Φ- and N Φ-approximants in Orlicz spaces of vector valued functions

Summary Let (, A, P) be a probability space and E be a Banach space. We study the approximation of an E-valued random variable X, which is an element of the Orlicz space L(, A, P; E), by a function YL, which is measurable with respect to a sub--field of A and takes values in a closed convex subset of E. Two types of approximation are considered: (X – Y) dP=inf, and N(X–Y)=inf with the Orlicz space norm N. We give conditions for the existence of best approximants. If E is reflexive, we obtain martingale type convergence theorems for best approximants and discuss the continuity of the operator X best approximant of X.This paper is a part of the authors doctoral thesis, written under the guidance of D. Landers     

Locally α-compact spaces based on continuous valued logic

This paper is a continuation of [1]. That is, it considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies), introduced by Ying [2]. It investigates topological notions defined by means of α-open sets when these are planted into the framework of Ying’s fuzzifying topological spaces (by Łukasiewicz logic in [0, 1]). Other characterizations of fuzzifying α-compactness are given, including characterizations in terms of nets and α-subbases. Several characterizations of locally α-compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained.     

Bishop–Phelps–Bollobás property for bilinear forms on spaces of continuous functions

It is shown that the Bishop–Phelps–Bollobás theorem holds for bilinear forms on the complex \(C_0(L_1)\times C_0(L_2)\) for arbitrary locally compact topological Hausdorff spaces \(L_1\) and \(L_2\).     

On Cesàro summability of vector valued multiplier spaces and operator valued series

In this paper, we introduce and study vector valued multiplier spaces with the help of the sequence of continuous linear operators between normed spaces and Cesàro convergence. Also, we obtain a new version of the Orlicz–Pettis Theorem by means of Cesàro summability.     

On the structure of sets of σ-monogeneity for continuous functions

The notion of the sets of -monogeneity for continuous functions is introduced which makes it possible to study pseudo-analytic properties of these functions. The theorem on the structure of these sets is proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 2, pp. 226–232, February, 1993.     

Refinements of the Cauchy–Bunyakovsky–Schwarz inequality for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous functions of selfadjoint operators in Hilbert spaces that improve the Cauchy–Bunyakovsky–Schwarz inequality, are given.     

Sequences of composition operators in spaces of functions of bounded Φ-variation

The main results of the paper are contained in Theorems 1 and 2. Theorem 1 presents necessary and sufficient conditions for a sequence of functions h _n : 〈c, d〉 → 〈a, b〉, n = 1, 2, ..., to have bounded sequences of Ψ-variations {V _Ψ (〈c, d〉; f ? h _n )} _n=1 ^∞ evaluated for the compositions of an arbitrary function f: 〈a, b〉 → ? with finite Φ-variation and the functions h _n . In Theorem 2, the same is done for a sequence of functions h _n : ? → ?, n = 1, 2, ..., and the sequence of Ψ-variations {V _Ψ(〈a, b〉; h _n ? f)} _n=1 ^∞ .     

Fixed point sets for abstract volterra operators on fréchet spaces

In this present article the topological of the solution ser for abstract Volterra equations is studied both in Banach spaces and in Fréchet spaces. It is shown that the solution set for certain nonlinear abstract Volterra equations in the Fréchet spaces C[0,∞) and L^p _loc[0,∞) (l≤p≤∞) are R_δ sets. Applications of the main results to nonlinear classical integral equations are given     

Sufficient sets in weighted Fréchet spaces of entire functions

Under study are sufficient sets in Fréchet spaces of entire functions with uniform weighted estimates. We obtain general results on the a priori overflow of these sets and introduce the concept of their minimality. We also establish necessary and sufficient conditions for a sequence of points on the complex plane to be a minimal sufficient set for a weighted Fréchet space. Applications are given to the problem of representation of holomorphic functions in a convex domain with certain growth near the boundary by exponential series.     

The spectrum is continuous on the set of p–hyponormal operators

In this paper it is shown that the spectrum , a set valued function, is continuous when the function is restricted to the set of all p-hyponormal operators on a Hilbert space. Received November 9, 1998; in final form August 6, 1999 / Published online July 3, 2000     

Riesz–Kantorovich formulas for operators on multi-wedged spaces

We introduce the notions of multi-suprema and multi-infima for vector spaces equipped with a collection of wedges, generalizing the notions of suprema and infima in ordered vector spaces. Multi-lattices are vector spaces that are closed under multi-suprema and multi-infima and are thus an abstraction of vector lattices. The Riesz decomposition property in the multi-wedged setting is also introduced, leading to Riesz–Kantorovich formulas for multi-suprema and multi-infima in certain spaces of operators.     

Generalised weighted composition operators on Maeda–Ogasawara spaces

Every Archimedean Riesz space can be embedded as an order dense subspace of some $C^{\infty }\left(X\right)$, the Riesz space of all extended continuous functions on a Stonean space $X$, called its Maeda–Ogasawara space. Furthermore, it is a fact that every Riesz homomorphism between spaces of ordinary continuous functions on compact Hausdorff spaces is a weighted composition operator. We prove that a generalised statement holds for Maeda–Ogasawara spaces and refine these results in case the homomorphism preserves order limits.     

Compactness and essential norms of composition operators on Orlicz–Lorentz spaces

In this work, we present necessary and sufficient conditions for compactness of the composition operator on Orlicz–Lorentz spaces and determine upper and lower estimates for the essential norm of the composition operator on these spaces.     

Singular integral operators on tent spaces over spaces of homogeneous type: Fefferman–Stein box maximal functions and pointwise Carleson type estimates

In this article we generalize the singular integral operator theory on weighted tent spaces to spaces of homogeneous type. This generalization of operator theory is in the spirit of C. Fefferman and Stein since we use some auxiliary functionals on tent spaces which play roles similar to the Fefferman–Stein sharp and box maximal functions in the Lebesgue space setting. Our contribution in this operator theory is twofold: for singular integral operators (including maximal regularity operators) on tent spaces pointwise Carleson type estimates are proved and this recovers known results; on the underlying space no extra geometrical conditions are needed and this could be useful for future applications to parabolic problems in rough settings.     

Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces

A boundedness result is established for multilinear singular integral operators on the homogeneous Morrey–Herz spaces. As applications, two corollaries about interesting cases of the boundedness of the considered operators on the homogeneous Morrey–Herz spaces are obtained.     

Boundedness of fractional integral operators on α-modulation spaces

In this paper, we obtain the boundedness of the fractional integral operators, the bilinear fractional integral operators and the bilinear Hilbert transform on α-modulation spaces.     

Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators

Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.