Vector measures with values in the compact operators. (English) Zbl 0645.46037

Ricker, Werner

Proc. Am. Math. Soc. 102, No. 2, 441-442 (1988).

A result of V. Kaftal and G. Weiss [Proc. Am. Math. Soc. 98, 431-435 (1986; Zbl 0616.47015)] on convergence of compact series of compact operators in Hilbert space has been extended and applied to the following result:
If m is a compact operator valued measure on a \(\sigma\)-algebra of sets, then it is \(\sigma\)-additive with respect to the uniform operator topology on B(H).
This result in turn simplifies the proof of a result of J. Diestel and B. Faires [Trans. Am. Math. Soc. 198, 253-271 (1974; Zbl 0297.46034)] on the uniform \(\sigma\)-additivity of integrals induced by compact selfadjoint operators.

Reviewer: P.Kruszynski

MSC:

46G10	Vector-valued measures and integration
47B06	Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
28B05	Vector-valued set functions, measures and integrals

Keywords:

convergence of compact series of compact operators; compact operator valued measure; uniform operator topology

Citations:

Zbl 0616.47015; Zbl 0297.46034