On positive operators on some ordered Banach spaces
A well-known theorem due to B.\,Maurey [cf.\ \textit{B.\,Maurey} and \textit{G.\,Pisier}, Stud.\ Math.\ 58, 45--90 (1976; Zbl 0344.47014)] asserts that if \(B\) is a Banach space having cotype \(q\geq2\) and \(K\) is a compact Hausdorff space, then each continuous linear map \(T:C(K)\rightarrow B\) is \(r\)-summing, for every \(r>q\). In the paper under review, among other results, the author obtains some variations (and a sketch of an alternative proof) of this theorem. The present author defines \(i(X)=\inf \left\{p\mid(\sum_{i=1}^{n}\left\| x_{i}\right\| ^{p})^{\frac{1}{p}}\leq C_{p}\left\| \sum_{i=1}^{n}x_{i}\right\| \right\}\), where \(x_{1},\dots,x_{n}\in X\), where \(X\) is a normal cone contained in a Banach space \(B\) and \(C_{p}\) is some constant. Then he presents an interesting counterpart of the result of Maurey, where \(i(X)\), in some sense, can be seen as a kind of cotype of the cone \(X\). More precisely, if \(K\) is a compact Hausdorff space and \(B\) is a Banach space, ordered by a normal cone \(X\) such that \(i(X)<\infty\), then, for \(r>i(X)\), each positive continuous operator \(T:C(K)\rightarrow B\) is \(r\)-summing and \(\Pi_{r}(T)\leq C\left\| T\right\|\), where \(C\) is some constant. Reciprocally, if \(r>1\) enjoys the preceding properties, then \(r\geq i(X)\). Some applications to conical measures are also presented. The last section of the paper is devoted to the study of positive operators with values in an \(L^{1}\)-space. These results can be regarded, in some sense, as a dual counterpart of the main result of the previous section.
scientific article; zbMATH DE number 5070756
scientific article