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Factorization through Hilbert space and the dilation of L(X,Y)-valued measures

V. Mandrekar, P. Richard (1993)

Studia Mathematica

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.

Fréchet-spaces-valued measures and the AL-property.

S. Okada, W. J. Ricker (2003)

RACSAM

Associated with every vector measure m taking its values in a Fréchet space X is the space L1(m) of all m-integrable functions. It turns out that L1(m) is always a Fréchet lattice. We show that possession of the AL-property for the lattice L1(m) has some remarkable consequences for both the underlying Fréchet space X and the integration operator f → ∫ f dm.

From weak to strong types of L E 1 -convergence by the Bocce criterion

Erik Balder, Maria Girardi, Vincent Jalby (1994)

Studia Mathematica

Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space ℒ E 1 to be norm convergent (resp. relatively norm compact), thus extending the known results for ℒ ℝ 1 . Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in ℒ E 1 . It is shown that tightness is a necessary and sufficient condition to pass from limited to strong convergence....

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