scholarly journals Isometric factorization of vector measures and applications to spaces of integrable functions

2021 ◽  
pp. 125857
Author(s):  
Olav Nygaard ◽  
José Rodríguez
Keyword(s):  
Vector Measures ◽  
Integrable Functions

scholarly journals Non-weak compactness of the integration map for vector measures

1993 ◽  
Vol 54 (3) ◽  
pp. 287-303 ◽  
Author(s):  
S. Okada ◽  
W. J. Ricker
Keyword(s):  
Banach Space ◽  
Volterra Operator ◽  
Continuous Operator ◽  
Weakly Compact ◽  
Vector Measures ◽  
Mean Convergence ◽  
Integrable Functions ◽  
The Mean ◽  
The Fourier Transform ◽  
Adjoint Operators

AbstractLet m be a vector measure with values in a Banach space X. If L1(m) denotes the space of all m integrable functions then, with respect to the mean convergence topology, L1(m) is a Banach space. A natural operator associated with m is its integration map Im which sends each f of L1(m) to the element ∫fdm (of X). Many properties of the (continuous) operator Im are closely related to the nature of the space L1(m). In general, it is difficult to identify L1(m). We aim to exhibit non-trivial examples of measures m in (non-reflexive) spaces X for which L1(m) can be explicitly computed and such that Im is not weakly compact. The examples include some well known operators from analysis (the Fourier transform on L1 ([−π, π]), the Volterra operator on L1 ([0, 1]), compact self-adjoint operators in a Hilbert space); such operators can be identified with integration maps Im (or their restrictions) for suitable measures m.

scholarly journals The Köthe Dual of an Abstract Banach Lattice

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
E. Jiménez Fernández ◽  
M. A. Juan ◽  
E. A. Sánchez-Pérez
Keyword(s):  
Integral Representation ◽  
Banach Lattice ◽  
Broad Class ◽  
Banach Lattices ◽  
Vector Measures ◽  
Integrable Functions ◽  
Suitable Definition ◽  
Integral Structures ◽  
Definition Of ◽  
Köthe Dual

We analyze a suitable definition of Köthe dual for spaces of integrable functions with respect to vector measures defined onδ-rings. This family represents a broad class of Banach lattices, and nowadays it seems to be the biggest class of spaces supported by integral structures, that is, the largest class in which an integral representation of some elements of the dual makes sense. In order to check the appropriateness of our definition, we analyze how far the coincidence of the Köthe dual with the topological dual is preserved.

Weak precompactness, strong boundedness, and weak complete continuity

1990 ◽  
Vol 108 (2) ◽  
pp. 325-335 ◽  
Author(s):  
Catherine A. Abbott ◽  
Elizabeth M. Bator ◽  
Russell G. Bilyeu ◽  
Paul W. Lewis
Keyword(s):  
Continuous Function ◽  
Uniform Integrability ◽  
Bounded Operators ◽  
Countable Additivity ◽  
Complete Continuity ◽  
Completely Continuous ◽  
Vector Measures ◽  
Integrable Functions ◽  
Completely Continuous Operators ◽  
Continuous Operators

AbstractWeak precompactness in spaces of vector measures and in the space of Bochner integrable functions is studied. Uniform countable additivity and uniform integrability are characterized in terms of weak precompactness. Through this, a connection between strongly bounded operators and operators having weakly precompact adjoints on abstract continuous function spaces is established. These operators are compared with weakly completely continuous operators.

scholarly journals Weak Compactness and Vector Measures

1955 ◽  
Vol 7 ◽  
pp. 289-305 ◽  
Author(s):  
R. G. Bartle ◽  
N. Dunford ◽  
J. Schwartz
Keyword(s):  
Banach Space ◽  
Spectral Theory ◽  
Weak Compactness ◽  
General Situation ◽  
Additive Measure ◽  
Convergence Theorems ◽  
Vector Measures ◽  
Ordinary Space ◽  
Integrable Functions ◽  
Recent Developments

Introduction. It is the purpose of this paper to develop a Lebesgue theory of integration of scalar functions with respect to a countably additive measure whose values lie in a Banach space. The class of integrable functions reduces to the ordinary space of Lebesgue integrable functions if the measure is scalar valued. Convergence theorems of the Vitali and Lebesgue type are valid in the general situation. The desirability of such a theory is indicated by recent developments in spectral theory.

scholarly journals Spaces of vector functions that are integrable with respect to vector measures

2007 ◽  
Vol 82 (1) ◽  
pp. 85-109 ◽  
Author(s):  
José Rodríguez
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Equivalence Classes ◽  
The Other ◽  
Normed Spaces ◽  
Vector Measures ◽  
Other Hand ◽  
Integrable Functions ◽  
Compact Range ◽  
Indefinite Integral

AbstractWe study the normed spaces of (equivalence classes of) Banach space-valued functions that are Dobrakov,S* or McShane integrable with respect to a Banach space-valued measure, where the norm is the natural one given by the total semivariation of the indefinite integral. We show that simple functions are dense in these spaces. As a consequence we characterize when the corresponding indefinite integrals have norm relatively compact range. On the other hand, we also determine when these spaces are ultrabornological. Our results apply to conclude, for instance, that the spaces of Birkhoff (respectively McShane) integrable functions defined on a complete (respectively quasi-Radon) probability space, endowed with the Pettis norm, are ultrabornological.

scholarly journals On Pointwise Product Vector Measure Duality

2021 ◽  
Vol 20 ◽  
pp. 8-18
Author(s):  
Levi Otanga Olwamba ◽  
Maurice Oduor
Keyword(s):  
Integral Representation ◽  
Function Space ◽  
Hilbert Function ◽  
Vector Measure ◽  
Inner Product ◽  
Vector Measures ◽  
Integrable Functions ◽  
Space Of Integrable Functions ◽  
Hilbert Function Space ◽  
Product Vector

This article is devoted to the study of pointwise product vector measure duality. The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral representation of product vector measures, inner product functions and products of measurable sets.

scholarly journals VECTOR MEASURES OF BOUNDED SEMIVARIATION AND ASSOCIATED CONVOLUTION OPERATORS

2010 ◽  
Vol 53 (2) ◽  
pp. 333-340
Author(s):  
PAULETTE SAAB ◽  
MANGATIANA A. ROBDERA
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Convolution Operators ◽  
Weakly Compact ◽  
Absolutely Continuous ◽  
Vector Measures ◽  
Injective Tensor Product ◽  
Integrable Functions ◽  
Compact Range ◽  
Space Of Integrable Functions

AbstractLet G be a compact metrizable abelian group, and let X be a Banach space. We characterize convolution operators associated with a regular Borel X-valued measure of bounded semivariation that are compact (resp; weakly compact) from L1(G), the space of integrable functions on G into L1(G) X, the injective tensor product of L1(G) and X. Along the way we prove a Fourier Convergence theorem for vector measures of relatively compact range that are absolutely continuous with respect to the Haar measure.

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