scholarly journals Weak Precompactness in the Space of Vector-Valued Measures of Bounded Variation

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Ioana Ghenciu
Keyword(s):  
Banach Space ◽  
Total Variation ◽  
Bounded Variation ◽  
Measure Space ◽  
Total Variation Norm ◽  
Variation Norm ◽  
Additive Measures ◽  
Vector Valued

For a Banach spaceXand a measure space(Ω,Σ), letM(Ω,X)be the space of allX-valued countably additive measures on(Ω,Σ)of bounded variation, with the total variation norm. In this paper we give a characterization of weakly precompact subsets ofM(Ω,X).

scholarly journals bvca(Σ, X) REVISITED

2010 ◽  
Vol 53 (2) ◽  
pp. 341-346
Author(s):  
J. C. FERRANDO
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Measurable Space ◽  
Sufficient Condition ◽  
General Sufficient Condition ◽  
Variation Norm ◽  
Additive Measures

AbstractAssuming that (Ω, Σ) is a measurable space and (X) is a Banach space we provide a quite general sufficient condition on (X) for bvca(Σ, X) (the Banach space of all X-valued countably additive measures of bounded variation equipped with the variation norm) to contain a copy of c0 if and only if X does. Some well-known results on this topic are straightforward consequences of our main theorem.

scholarly journals Complex extreme measurable selections

1995 ◽  
Vol 58 (2) ◽  
pp. 222-231
Author(s):  
Douglas Mupasiri
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Extreme Point ◽  
Measure Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Closed Unit Ball ◽  
Measurable Selections ◽  
Necessary And Sufficient

AbstractWe give a characterization of complex extreme measurable selections for a suitable set-valued map. We use this result to obtain necessary and sufficient conditions for a function to be a complex extreme point of the closed unit ball of Lp (ω, Σ, ν X), where (ω, σ, ν) is any positive, complete measure space, X is a separable complex Banach space, and 0 < p < ∞.

scholarly journals Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

scholarly journals Existence and Uniqueness of a Quasistationary Distribution for Markov Processes with Fast Return from Infinity

2014 ◽  
Vol 51 (3) ◽  
pp. 756-768 ◽  
Author(s):  
Servet Martínez ◽  
Jaime San Martín ◽  
Denis Villemonais
Keyword(s):  
Total Variation ◽  
Existence And Uniqueness ◽  
Initial Distribution ◽  
Total Variation Norm ◽  
Birth And Death Processes ◽  
Long Time Behaviour ◽  
Long Time ◽  
Unique Distribution ◽  
Variation Norm ◽  
Quasistationary Distributions

We study the long-time behaviour of a Markov process evolving in N and conditioned not to hit 0. Assuming that the process comes back quickly from ∞, we prove that the process admits a unique quasistationary distribution (in particular, the distribution of the conditioned process admits a limit when time goes to ∞). Moreover, we prove that the distribution of the process converges exponentially fast in the total variation norm to its quasistationary distribution and we provide a bound for the rate of convergence. As a first application of our result, we bring a new insight on the speed of convergence to the quasistationary distribution for birth-and-death processes: we prove that starting from any initial distribution the conditional probability converges in law to a unique distribution ρ supported in N* if and only if the process has a unique quasistationary distribution. Moreover, ρ is this unique quasistationary distribution and the convergence is shown to be exponentially fast in the total variation norm. Also, considering the lack of results on quasistationary distributions for nonirreducible processes on countable spaces, we show, as a second application of our result, the existence and uniqueness of a quasistationary distribution for a class of possibly nonirreducible processes.

scholarly journals Weak compactness in spaces of vector valued measures

1988 ◽  
Vol 38 (1) ◽  
pp. 55-56
Author(s):  
F.G.J. Wiid
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Weak Compactness ◽  
Vector Valued

We characterise relative weak compactness in σBM(∑, X), the space of sigma-additive, X-valued measures of bounded variation, where X is a Banach space.

Characterization of some classes of operators on spaces of vector-valued continuous functions

1985 ◽  
Vol 97 (1) ◽  
pp. 137-146 ◽  
Author(s):  
Fernando Bombal ◽  
Pilar Cembranos
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Representing Measure ◽  
Bounded Linear ◽  
Second Dual ◽  
Image Position ◽  
Vector Valued

Let K be a compact Hausdorff space and E, F Banach spaces. We denote by C(K, E) the Banach space of all continuous. E-valued functions defined on K, with the supremum norm. It is well known ([6], [7]) that every operator (= bounded linear operator) T from C(K, E) to F has a finitely additive representing measure m of bounded semi-variation, defined on the Borel σ-field Σ of K and with values in L(E, F″) (the space of all operators from E into the second dual of F), in such a way thatwhere the integral is considered in Dinculeanu's sense.

scholarly journals Total Variation Applications in Computer Vision

2018 ◽  
pp. 543-570
Author(s):  
Vania Vieira Estrela ◽  
Hermes Aguiar Magalhães ◽  
Osamu Saotome
Keyword(s):  
Computer Vision ◽  
Total Variation ◽  
Total Variation Norm ◽  
Mathematical Background ◽  
Tv Regularization ◽  
Variation Norm ◽  
Image Processing Algorithms ◽  
Processing Algorithms ◽  
Set Up

The objectives of this chapter are: (i) to introduce a concise overview of regularization; (ii) to define and to explain the role of a particular type of regularization called total variation norm (TV-norm) in computer vision tasks; (iii) to set up a brief discussion on the mathematical background of TV methods; and (iv) to establish a relationship between models and a few existing methods to solve problems cast as TV-norm. For the most part, image-processing algorithms blur the edges of the estimated images, however TV regularization preserves the edges with no prior information on the observed and the original images. The regularization scalar parameter λ controls the amount of regularization allowed and it is essential to obtain a high-quality regularized output. A wide-ranging review of several ways to put into practice TV regularization as well as its advantages and limitations are discussed.

Characterization of a Class of Equicontinuous Sets of Finitely Additive Measures with an Application to Vector Valued Borel Measures

1974 ◽  
Vol 26 (02) ◽  
pp. 281-290 ◽  
Author(s):  
Richard Alan Oberle
Keyword(s):  
Abstract Space ◽  
Borel Measures ◽  
Disjoint Sequence ◽  
Finitely Additive Measures ◽  
Additive Measures ◽  
Positive Integers ◽  
Vector Valued ◽  
Increasing Functions

Let V denote a ring of subsets of an abstract space X, let R + denote the nonnegative reals, and let N denote the set of positive integers. We denote by C(V) the space of all subadditive and increasing functions, from the ring V into R +, which are zero at the empty set. The space C(V) is called the space of contents on the ring V and elements are referred to as contents. A sequence of sets An ∊ V, n ∊ N is said to be dominated if there exists a set B ∊ V such that An ⊆ B, for n = 1, 2, A content p ∊ C(V) is said to be Rickart on the ring V if lim n p(An ) = 0 for each dominated, disjoint sequence An ∊ V, n ∊ N.

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