On some open problems involving range of vector measures (original) (raw)
On Some Aspects of Vector Measures
Advances in Mathematics: Scientific Journal
This paper addresses some properties of vector measures (Banach space valued measures) as well as topological results on some spaces of vector measures of bounded variation.
Remarks on the semivariation of vector measures with respect to Banach spaces
Bulletin of the Australian Mathematical Society, 2007
Suppose that and . It is shown that any Lp(µ)-valued measure has finite L2(v)-semivariation with respect to the tensor norm for 1 ≤ p < ∞ and finite Lq(v)-semivariation with respect to the tensor norm whenever either q = 2 and 1 ≤ p ≤ 2 or q > max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor norm for any 1 ≤ q < 2. It is also shown that the measure m (A) = χA has infinite Lq-semivariation with respect to the tensor norm if q < p.
Studia Mathematica, 2014
It is proved that if X is infinite-dimensional, then there exists an infinitedimensional space of X-valued measures which have infinite variation on sets of positive Lebesgue measure. In term of spaceability, it is also shown that ca(B, λ, X) \ Mσ, the measures with non-σ-finite variation, contains a closed subspace. Other considerations concern the space of vector measures whose range is neither closed nor convex. All of those results extend in some sense theorems of Muñoz Fernández et al. [Linear Algebra Appl. 428 (2008)].
The Vector Measures Whose Range Is Strictly Convex
Journal of Mathematical Analysis and Applications, 1999
In this paper we resume the most important results that we obtained in our papers [1,2,5,6,7] concerning a broad class of measures that we defined in dealing with a bangbang control problem. Let M be the σ−algebra of the Lebesgue measurable subsets of [0, 1] and µ : M → R n be a non-atomic vector measure. A well known Theorem of Lyapunov (see [11]) states that the range of µ, defined by R(µ) = {µ(E) : E ∈ M}, is closed and convex or, equivalently, that given a measurable function ρ with values in [0, 1] there exists a set E in M such that (*) X ρ dµ = µ(E). Lyapunov's Theorem has been widely applied in bang-bang control theory [10] and, more recently, in some non-convex problems of the Calculus of Variations [3]. As an example we mention the following bang-bang existence result:
VECTOR MEASURES WITH VARIATION IN A BANACH FUNCTION SPACE
Function Spaces - Proceedings of the Sixth Conference, 2003
Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the Köthe-Bochner function space defined as the set of measurable functions f : Ω → X such that the nonnegative functions f X : Ω → [0, ∞) are in the lattice E. The notion of E-variation of a measure -which allows to recover the pvariation (for E = L p ), Φ-variation (for E = L Φ ) and the general notion introduced by Gresky and Uhl-is introduced. The space of measures of bounded E-variation V E (X) is then studied. It is shown, among other things and with some restriction of absolute continuity of the norms, that (E(X)) * = V E (X * ), that V E (X) can be identified with space of cone absolutely summing operators from E into X and that E(X) = V E (X) if and only if X has the RNP property.
Vector measures: where are their integrals?
Positivity, 2008
Let ν be a vector measure with values in a Banach space Z. The integration map Iν : L 1 (ν) → Z, given by f → f dν for f ∈ L 1 (ν), always has a formal extension to its bidual operator I * * ν : L 1 (ν) * * → Z * *. So, we may consider the "integral" of any element f * * of L 1 (ν) * * as I * * ν (f * *). Our aim is to identify when these integrals lie in more tractable subspaces Y of Z * *. For Z a Banach function space X, we consider this question when Y is any one of the subspaces of X * * given by the corresponding identifications of X, X (the Köthe bidual of X) and X * (the topological dual of the Köthe dual of X). Also, we consider certain kernel operators T and study the extended operator I * * ν for the particular vector measure ν defined by ν(A) := T (χA).
A Note on Radon-Nikodym Theorem for Operator Valued Measures and Its Applications
Communications of the Korean Mathematical Society
In this note we present sufficient conditions for the existence of Radon-Nikodym derivatives (RND) of operator valued measures with respect to scalar measures. The RND is characterized by the Bochner integral in the strong operator topology of a strongly measurable operator valued function with respect to a nonnegative finite measure. Using this result we also obtain a characterization of compact sets in the space of operator valued measures. An extension of this result is also given using the theory of Pettis integral. These results have interesting applications in the study of evolution equations on Banach spaces driven by operator valued measures as structural controls.
Open Problems in the Geometry and Analysis of Banach Spaces
2016
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.