On vector measures and extensions of transfunctions

arXiv (Cornell University), 2017

If µ1, µ2, . . . are positive measures on a measurable space (X, Σ) and v1, v2, . . . are elements of a Banach space E such that ∞ n=1 vn µn(X) < ∞, then ω(S) = ∞ n=1 vnµn(S) defines a vector measure of bounded variation on (X, Σ). We show E has the Radon-Nikodym property if and only if every E-valued measure of bounded variation on (X, Σ) is of this form. As an application of this result we show that under natural conditions an operator defined on positive measures, has a unique extension to an operator defined on Evalued measures for any Banach space E that has the Radon-Nikodym property.

Arkiv för Matematik, 2010

A moment problem is presented for a class of signed measures which are termed pseudo-positive. Our main result says that for every pseudopositive definite functional (subject to some reasonable restrictions) there exists a representing pseudo-positive measure.

2005

In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVM's into POVM's, generally irreversibly, thus loosing some of the information retrieved from the measurement. This poses the problem of which POVM's are "undisturbed", namely they are not irreversibly connected to another POVM. We will call such POVM's clean. In a sense, the clean POVM's would be "perfect", since they would not have any additional "extrinsical" noise. Quite unexpectedly, it turns out that such cleanness property is largely unrelated to the convex structure of POVM's, and there are clean POVM's that are not extremal and vice-versa. In this paper we solve the cleannes classification problem for number n of outcomes n ≤ d (d dimension of the Hilbert space), and we provide a a set of either necessary or sufficient conditions for n > d, along with an iff condition for the case of informationally complete POVM's for n = d 2 .

Filomat, 2012

The close connection between the geometry of a Banach space and the properties of vector measures acting into it is now fairly well-understood. The present paper is devoted to a discussion of some of these developments and certain problems arising naturally in this circle of ideas which are either open or have been partially resolved. Emphasis shall be laid mainly on those aspects of this theory which involve properties of the range of these 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.

1987

The proof of a result analogous to that in Koelman and de Muynck [Phys. Lett. A 98, 1 (1983)] is given for the case of unbounded observables. If two, not necessarily bounded, observables are represented by a positive operator-valued measure, then the measurement of any of them is undisturbed if and only if they commute. The Naimark theorem on dilations of spectral functions is exploited. A stronger version of Wigner's theorem is given.

2003

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Colloquium Mathematicum, 2015

For each vector v we define the notion of a v-positive type for infinite measure-preserving transformations, a refinement of positive type as introduced by Hajian and Kakutani. We prove that a positive type transformation need not be (1, 2)-positive type. We study this notion in the context of Markov shifts and multiple recurrence and give several examples.

Acta Applicandae Mathematicae, 2019

Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure F : Ω → B(H ) has an integral representation of the form for some weakly measurable maps G k (1 ≤ k ≤ m) from a measurable space Ω to a Hilbert space H and some positive measure μ on Ω. Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic.

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE

We characterize all (signed) measures in BV n n−1 (R n) * , where BV n n−1 (R n) is defined as the space of all functions u in L n n−1 (R n) such that Du is a finite vector-valued measure. We also show that BV n n−1 (R n) * and BV (R n) * are isometrically isomorphic, where BV (R n) is defined as the space of all functions u in L 1 (R n) such that Du is a finite vector-valued measure. As a consequence of our characterizations, an old issue raised in Meyers-Ziemer [16] is resolved by constructing a locally integrable function f such that f belongs to BV (R n) * but |f | does not. Moreover, we show that the measures in BV n n−1 (R n) * coincide with the measures inẆ 1,1 (R n) * , the dual of the homogeneous Sobolev spaceẆ 1,1 (R n), in the sense of isometric isomorphism. For a bounded open set Ω with Lipschitz boundary, we characterize the measures in the dual space BV 0 (Ω) *. One of the goals of this paper is to make precise the definition of BV 0 (Ω), which is the space of functions of bounded variation with zero trace on the boundary of Ω. We show that the measures in BV 0 (Ω) * coincide with the measures in W 1,1 0 (Ω) *. Finally, the class of finite measures in BV (Ω) * is also characterized.