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Factorization through Hilbert space and the dilation of $$L(X,Y)$$-valued measures. (English) Zbl 0812.47016
Summary: We present a general necessary and sufficient algebraic condition for the spectral dilation of a finitely additive $$L(X,Y)$$-valued measure of finite semivariation when $$X$$ and $$Y$$ are Banach spaces. Using our condition we derive the main results of Rosenberg, Makagon and Salehi, and Miamee without the assumption that $$X$$ and/or $$Y$$ are Hilbert spaces. In addition we relate the dilation problem to the problem of factoring a family of operators through a single Hilbert space.
##### MSC:
 47A66 Quasitriangular and nonquasitriangular, quasidiagonal and nonquasidiagonal linear operators 47A20 Dilations, extensions, compressions of linear operators 46G10 Vector-valued measures and integration 60B11 Probability theory on linear topological spaces 28B05 Vector-valued set functions, measures and integrals
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