Probabilistic norms for linear operators
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.
scientific article; zbMATH DE number 1218573
scientific article