Bounded sets in projective tensor products of hilbertizable locally convex spaces

Klaus-Detlef Kürsten

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The problem of topologies of Grothendieck is considered for complete tensor products of Fréchet spaces endowed with the topology defined by an arbitrary tensor norm. Some consequences on the stability of certain locally convex properties in spaces of operators are also given.

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The space of the fully absolutely (r;r1,...,rn)-summing n-linear mappings between Banach spaces is introduced along with a natural (quasi-)norm on it. If r,rk C [1,+infinite], k=1,...,n, this space is characterized as the topological dual of a space of virtually nuclear mappings. Other examples and properties are considered and a relationship with a topological tensor product is stablished. For Hilbert spaces and r = r1 = ... = rn C [2,+infinite[ this space is isomorphic to the space...

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We study tensor norms and operator ideals related to the ideal $P_{p, σ}$ , 1 < p < ∞, 0 < σ < 1, of (p,σ)-absolutely continuous operators of Matter. If α is the tensor norm associated with $P_{p, σ}$ (in the sense of Defant and Floret), we characterize the ${(α^{'})}^{t}$ -nuclear and ${(α^{'})}^{t}$ - integral operators by factorizations by means of the composition of the inclusion map $L^{r} (μ) \to L^{1} (μ) + L^{p} (μ)$ with a diagonal operator $B_{w} : L^{\infty} (μ) \to L^{r} (μ)$ , where r is the conjugate exponent of p’/(1-σ). As an application we study the reflexivity of the components...