Probabilistic norms for linear operators
type: Item
instance of: scholarly article
title: Probabilistic norms for linear operators
zbMATH Open document ID: 0922.47068
author: Carlo Sempi
author: Bernardo Lafuerza Guillén
author: José Antonio Rodríguez-Lallena
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-11-03T00:00:00Z
review text: Let \(V_1\) and \(V_2\) be probabilistic normed (PN) spaces, and \(L\) the space of all linear operators \(T: V_1\to V_2\). The authors study the following subsets of \(L\): \(L_b\) probabilistic bounded operators, \(L_c\) continuous operators and \(L_{bc}= L_b\cap L_c\). They work with the Sibley metric on the space of distribution functions. Given a subset \(A\subset V_1\), for each operator \(T\in L\) one can define the distribution function \(\nu^A(T)\) as the probabilistic radius of \(T(A)\). One of the main results of this paper, Theorem 3.1, says that \(\nu^A\) is a probabilistic pseudonorm on \(L\), and the convergence in \(\nu^A\) is equivalent to the uniform convergence on \(A\). This theorem and its corollaries generalize and strengthen the results of \textit{V. Radu} [C. R. Acad. Sci., Paris, Sér. A 280, 1303-1305 (1975)]. Then the authors give different characterizations of the classes \(L_c\), \(L_b\) and \(L_{bc}\), and study when the corresponding PN spaces of operators are complete. The final part of the paper is devoted to equicontinuous and uniformly bounded families of operators.
Mathematics Subject Classification ID: 47S50
Mathematics Subject Classification ID: 54E70
zbMATH DE Number: 1218573
zbMATH Keywords: probabilistic normed spaces
zbMATH Keywords: probabilistic bounded operators
zbMATH Keywords: Sibley metric
zbMATH Keywords: space of distribution functions
zbMATH Keywords: PN spaces
reviewed by: Andrzej Nowak
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5810
OpenAlex ID: W2060518928
cites work: On the definition of a probabilistic normed space
cites work: Continuity properties of probabilistic norms
cites work: Q4151425
cites work: Q4360868
cites work: A study of boundedness in probabilistic normed spaces
cites work: Completion of probabilistic normed spaces
cites work: Equicontinuity, Affine Mean Ergodic Theorem and Linear Equations in Random Normed Spaces
cites work: Multiplications on the space of probability distribution functions
cites work: Q3336431
cites work: Q3664117
cites work: A metric for weak convergence of distribution functions
cites work: Q3249198
cites work: Q3746565
DOI: 10.1006/JMAA.1997.5810
Recommended article: Q5289541
Recommended article: On probabilistic norms of linear operators and spaces of operators
Recommended article: Linear operators in finite dimensional probabilistic normed spaces
Recommended article: Q3771309
Recommended article: Probabilistic norm of linear operators on PN space
Recommended article: Q4412360
Recommended article: Q3313052
Recommended article: Q3527425
Recommended article: Q4290540
Recommended article: Q4603308
description: scientific article; zbMATH DE number 1218573