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Probabilistic norms for linear operators. (English) Zbl 0922.47068
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 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.
Reviewer: A.Nowak (Katowice)

##### MSC:
 47S50 Operator theory in probabilistic metric linear spaces 54E70 Probabilistic metric spaces
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##### References:
 [1] Alsina, C.; Schweizer, B.; Sklar, A., On the definition of a probabilistic normed space, Aequationes math., 46, 91-98, (1993) · Zbl 0792.46062 [2] Alsina, C.; Schweizer, B.; Sklar, A., Continuity properties of probabilistic norms, J. math. anal. appl., 208, 446-452, (1997) · Zbl 0903.46075 [3] Bocşan, G., Sur certaines semi-normes aléatoires et leurs applications, C.R. acad. sci. Paris Sér. A, 282, 1319-1321, (1976) · Zbl 0374.60008 [4] Lafuerza Guillén, B.; Rodriguez Lallena, J.A.; Sempi, C., Some classes of probabilistic normed spaces, Rend. mat. (7), 17, 237-252, (1997) · Zbl 0904.54025 [5] B. Lafuerza Guillén, J. A. Rodriguez Lallena, C. Sempi, A study of boundedness in probabilistic normed spaces · Zbl 0945.46056 [6] Lafuerza Guillén, B.; Rodriguez Lallena, J.A.; Sempi, C., Completion of probabilistic normed spaces, Internat. J. math. math. sci., 18, 649-652, (1995) · Zbl 0855.54042 [7] Radu, V., Sur une norme aléatoire et la continuité des opérations linéaires dans des espaces normés aléatoires, C. R. acad. sci. Paris Sér. A, 280, 1303-1305, (1975) [8] Radu, V., Equicontinuity, affine Mean erogodic theorem and linear equations in random normed spaces, Proc. amer. math. soc., 57, 299-303, (1976) · Zbl 0348.47008 [9] Schweizer, B., Multiplication on the space of probability distribution functions, Aequationes math., 12, 156-183, (1975) · Zbl 0305.22004 [10] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), Elsevier, North-Holland New York · Zbl 0546.60010 [11] Sempi, C., On the space of distribution functions, Riv. mat. univ. parma (4), 8, 243-250, (1982) · Zbl 0516.60016 [12] Sibley, D.A., A metric for weak convergence of distribution functions, Rocky mountain J. math., 1, 427-430, (1971) · Zbl 0245.60007 [13] Taylor, A.E., Introduction to functional analysis, (1958), Wiley New York · Zbl 0081.10202 [14] Taylor, M.D., New metrics for weak convergence of distribution functions, Stochastica, 9, 5-17, (1985) · Zbl 0607.60004
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