Swartz’s Orlicz-Pettis theorem for operators. (English) Zbl 0947.46001

Let \(E\) and \(F\) be normed spaces and \(L(E,F)\) be the space of all continuous linear operators from \(E\) into \(F\). C. W. Swartz introduced in [Southeast Asian Bull. Math. 12, No. 1, 31-38 (1988; Zbl 0711.47024)] a topology \(L_A(E,F)\) on \(L(E,F)\), such that for each series \(\sum_jT_j\) in \(L(E,F)\), if \(\sum_jT_j\) is subseries (s.s.) convergent in the weak operator topology, then \(\sum_jT_j\) is s.s. convergent in \(L_A(E,F)\) and so he established an Orlicz-Pettis theorem for operators. In the paper under review the authors construct a new topology on \(L(E,F)\), which they denote by \(L_{M_0}(E,F)\) and they show that it is, in a certain sense, the strongest locally convex topology on \(L(E,F)\) having the mentioned property.


46A03 General theory of locally convex spaces
46B28 Spaces of operators; tensor products; approximation properties


Zbl 0711.47024