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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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