A characterization of absolutely summing operators by means of McShane integrable functions

Absolutely summing operators between Banach spaces are characterized by means of McShane integrable functions.     

向量值函数的Mcshane积分

本文讨论了向量值函数的Ｍｃｓｈａｎｅ积分的性质．例如收敛定理和原函数的性质等．     

Abstract extrapolation theorems for absolutely summing nonlinear operators

In this paper we prove a general version of the extrapolation theorem for absolutely summing nonlinear operators. It is explicitly shown that this result encompasses the previous old and recent, linear and nonlinear extrapolation theorems as particular cases. One of the steps of the proof provides another nonlinear extrapolation theorem of independent interest.     

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

We show that McShane and Pettis integrability coincide for functions , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function and an absolutely summing operator u from X to another Banach space Y such that the composition is not Bochner integrable; in particular, h is not McShane integrable.     

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

We show that McShane and Pettis integrability coincide for functions , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function and an absolutely summing operator u from X to another Banach space Y such that the composition is not Bochner integrable; in particular, h is not McShane integrable.     

Henstock-Kurzweil and McShane product integration; Descriptive definitions

The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function A is absolutely continuous. As a consequence we obtain that the McShane product integral of A over [a, b] exists and is invertible if and only if A is Bochner integrable on [a, b]. Supported by grant No. 201/04/0690 of the Grant Agency of the Czech Republic.     

On Henstock–Kurzweil and McShane integrals of Banach space-valued functions

This paper deals with the relation between the McShane integral and the Henstock–Kurzweil integral for the functions mapping a compact interval into a Banach space X and some other questions in connection with the McShane integral and the Henstock–Kurzweil integral of Banach space-valued functions. We prove that if a Banach space-valued function f is Henstock–Kurzweil integrable on I₀ and satisfies Property (P), then I₀ can be written as a countable union of closed sets E_n such that f is McShane integrable on each E_n when X contains no copy of c₀. We further give an answer to the Karták's question.     

On integration of vector functions with respect to vector measures

We study integration of Banach space-valued functions with respect to Banach space-valued measures. We focus our attention on natural extensions to this setting of the Birkhoff and McShane integrals. The corresponding generalization of the Birkhoff integral was first considered by Dobrakov under the name S*-integral. Our main result states that S*-integrability implies McShane integrability in contexts in which the later notion is definable. We also show that a function is measurable and McShane integrable if and only if it is Dobrakov integrable (i.e. Bartle *-integrable).     

A characterization of strongly measurable Kurzweil–Henstock integrable functions and weakly continuous operators

We give necessary and sufficient conditions for the Kurzweil–Henstock integrability of functions given by , where x_n belong to a Banach space and the sets (E_n)_n are measurable and pairwise disjoint. Also weakly completely continuous operators between Banach spaces are characterized by means of scalarly Kurzweil–Henstock integrable functions.     

An algebra of absolutely continuous functions and its multipliers

The aim of this paper is to study the algebraAC _p of absolutely continuous functionsf on [0,1] satisfying f(0) = 0,f ’ ∈ L^p[0, 1] and the multipliers ofAC _p .     

Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

In the smooth scattering theory framework, we consider a pair of self-adjoint operators H₀, H and discuss the spectral projections of these operators corresponding to the interval (−∞,λ). The purpose of the paper is to study the spectral properties of the difference D(λ) of these spectral projections. We completely describe the absolutely continuous spectrum of the operator D(λ) in terms of the eigenvalues of the scattering matrix S(λ) for the operators H₀ and H. We also prove that the singular continuous spectrum of the operator D(λ) is empty and that its eigenvalues may accumulate only at “thresholds” in the absolutely continuous spectrum.     

Homomorphisms and composition operators on algebras of analytic functions of bounded type

Let U and V be convex and balanced open subsets of the Banach spaces X and Y, respectively. In this paper we study the following question: given two Fréchet algebras of holomorphic functions of bounded type on U and V, respectively, that are algebra isomorphic, can we deduce that X and Y (or X^* and Y^*) are isomorphic? We prove that if X^* or Y^* has the approximation property and H_wu(U) and H_wu(V) are topologically algebra isomorphic, then X^* and Y^* are isomorphic (the converse being true when U and V are the whole space). We get analogous results for H_b(U) and H_b(V), giving conditions under which an algebra isomorphism between H_b(X) and H_b(Y) is equivalent to an isomorphism between X^* and Y^*. We also obtain characterizations of different algebra homomorphisms as composition operators, study the structure of the spectrum of the algebras under consideration and show the existence of homomorphisms on H_b(X) with pathological behaviors.     

The McShane, PU and Henstock integrals of Banach valued functions

Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals of vector valued functions are characterized.     

Approximation of functions by certain nonlinear integral operators

We study approximation properties of certain nonlinear integral operators L _n * obtained by a modification of given operators L _n . The operators L _n;r and L _n;r * of r-times differentiable functions are also studied. We give theorems on approximation orders of functions by these operators in polynomial weight spaces.     

Boundedness of the anisotropic maximal and anisotropic singular integral operators in generalized Morrey spaces

In this paper we give the conditions on the pair (ω ₁, ω ₂) which ensures the boundedness of the anisotropic maximal operator and anisotropic singular integral operators from one generalized Morrey space M_p,w1 \mathcal{M}_{p,\omega _1 } to another M_p,w2 \mathcal{M}_{p,\omega _2 }, 1 < p < g8, and from the space M_1,w1 \mathcal{M}_{1,\omega _1 } to the weak space WM_1,w2 W\mathcal{M}_{1,\omega _2 }.     

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.