On the square root of a positive B(*)-valued function
Let ([Omega], , [mu]) be a measure space, a separable Banach space, and * the space of all bounded conjugate linear functionals on . Let f be a weak* summable positive B(*)-valued function defined on [Omega]. The existence of a separable Hilbert space , a weakly measurable B()-valued function Q satisfying the relation Q*([omega])Q([omega]) = f([omega]) is proved. This result is used to define the Hilbert space L2,f of square integrable operator-valued functions with respect to f. It is shown that for B+(*)-valued measures, the concepts of weak*, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.
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Volume (Year): 7 (1977)
Issue (Month): 4 (December)
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