Absolutely summing mappings of locally convex spaces in measure theory. (Q1432653)

In this paper, some characterizations of absolutely summing operators on Hilbert spaces \(H\) with values in locally convex spaces \(X\) are given. For example: An operator \(S: H \rightarrow X\) is absolutely summing iff it factorizes through a Hilbert-Schmidt mapping. This result follows easily from the Pietsch factorization theorem and the result due to Pietsch from 1972 that each operator \(L^2\rightarrow C(K) \rightarrow L^2(K)\) is a Hilbert-Schmidt operator. As an application, it is shown that an operator \(S:H \rightarrow X\) is radonifying iff it is absolutely summing. This is a special case of a result of \textit{W. Linde} [Proc. Semin. Random Series, Convex Sets, Geom. Banach Spaces; Aarhus 1974, 127--133 (1975; Zbl 0327.46029)] stating that such an equivalence holds true on all Banach spaces \(H\) with the Radon-Nikodym property.