On the structure of a vector measure.
On vector lattice-valued measures. I.
On vector measures
Let be the Banach space of real measures on a -ring , let be its dual, let be a quasi-complete locally convex space, let be its dual, and let be an -valued measure on . If is shown that for any there exists an element of such that for any and that the mapis order continuous. It follows that the closed convex hull of is weakly compact.
On Vector Measures in Cartesian Products
On vector measures which have everywhere infinite variation or noncompact range [Book]
On vector sums of measure zero sets.
On Weak Compactness and Lower Closure Results for Pettis Integrable (Multi)Functions
In [4, 5, 7] an abstract, versatile approach was given to sequential weak compactness and lower closure results for scalarly integrable functions and multifunctions. Its main tool is an abstract version of the Komlós theorem, which applies to scalarly integrable functions. Here it is shown that this same approach also applies to Pettis integrable multifunctions, because the abstract Komlós theorem can easily be extended so as to apply to generalized Pettis integrable functions. Some results in the...
On weakly measurable stochastic processes and absolutely summing operators
A characterization of absolutely summing operators by means of McShane integrable stochastic processes is considered.
Operator ideal properties of vector measures with finite variation
Given a vector measure m with values in a Banach space X, a desirable property (when available) of the associated Banach function space L¹(m) of all m-integrable functions is that L¹(m) = L¹(|m|), where |m| is the [0,∞]-valued variation measure of m. Closely connected to m is its X-valued integration map Iₘ: f ↦ ∫f dm for f ∈ L¹(m). Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property L¹(m) = L¹(|m|) and the...
Operators from C*-algebras to Banach Spaces.
Operator-Valued Positive Definite Kernels on Tubes.
Optimal domains for the kernel operator associated with Sobolev's inequality
Refinements of the classical Sobolev inequality lead to optimal domain problems in a natural way. This is made precise in recent work of Edmunds, Kerman and Pick; the fundamental technique is to prove that the (generalized) Sobolev inequality is equivalent to the boundedness of an associated kernel operator on [0,1]. We make a detailed study of both the optimal domain, providing various characterizations of it, and of properties of the kernel operator when it is extended to act in its optimal domain....
Orthogonal Expansion of Vector-Valued Functions and Measures
p-integrale selectors of multimeasures.
Points fixes et théorèmes ergodiques dans les espaces L¹(E)
We prove that for each linear contraction T : X → X (∥T∥ ≤ 1), the subspace F = {x ∈ X : Tx = x} of fixed points is 1-complemented, where X is a suitable subspace of L¹(E*) and E* is a separable dual space such that the weak and weak* topologies coincide on the unit sphere. We also prove some related fixed point results.
Pointwise compactness and continuity of the integral.
In this paper we bring together the different known ways of establishing the continuity of the integral over a uniformly integrable set of functions endowed with the topology of pointwise convergence. We use these techniques to study Pettis integrability, as well as compactness in C(K) spaces endowed with the topology of pointwise convergence on a dense subset D in K.
Pointwise Compactness and Weakly Compact Sets in L...
Positive operator bimeasures and a noncommutative generalization
For C*-algebras A and B and a Hilbert space H, a class of bilinear maps Φ: A× B → L(H), analogous to completely positive linear maps, is studied. A Stinespring type representation theorem is proved, and in case A and B are commutative, the class is shown to coincide with that of positive bilinear maps. As an application, the extendibility of a positive operator bimeasure to a positive operator measure is shown to be equivalent to various conditions involving positive scalar bimeasures, pairs of...
Positive vector measures with given marginals
Suppose is an ordered locally convex space, and Hausdorff completely regular spaces and a uniformly bounded, convex and closed subset of . For , let . Then, under some topological and order conditions on , necessary and sufficient conditions are established for the existence of an element in , having marginals and .
Product of dominated vector measures