Swartz type results for nuclear and multiple 1-summing bilinear operators on \(c_0(\mathcal {X})\times c_0(\mathcal {Y})\)
In the paper under review the authors present a certain ``natural'' bilinear versions of \textit{Ch. Swartz}'s theorem [Trans. Am. Math. Soc., 179, 123--132 (1973; Zbl 0226.46038)] for bilinear operators on \(c_0(\mathcal{X})\), where \({\mathcal X} = (X_n)_{n=1}^\infty\) is a sequence of Banach spaces, and focus their attention to the validity of this theorem for nuclear and multiple \(1\)-summing bilinear operators. First they give some characterizations and examples of certain types of nuclear and multiple \(1\)-summing bilinear operators on these spaces, emphasizing the different behavior of the two classes. Here it is worth mentioning that, on \(c_0\), the class of nuclear operators coincide with those being multiple \(1\)-summing. They prove this is not true on \(c_0(\mathcal{X})\). Finally it is shown that this bilinear version of the Swartz's theorem fails for multiple \(1\)-summing bilinear operators.
scientific article; zbMATH DE number 6484280
scientific article