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

Junde, W.; Pap, E.; Li, Ronglu; Wenbo, Q.

Bull., Cl. Sci. Math. Nat., Sci. Math. 114, No. 22, 37-44 (1997).

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.

Reviewer: Zoran Kadelburg (Zemun)

MSC:

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

Keywords:

subseries convergence; weak operator topology; Orlicz-Pettis theorem

Citations:

Zbl 0711.47024