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Subspaces of \(L_p\) for \(0 \leq p < 1\) that are admissible as kernels. (English) Zbl 1038.46003

For \(L_p=L_p[0,1]\) spaces, \(0\leqslant p<1\), the author establishes a condition for a number of subspaces to be admissible as kernels (meaning that such a subspace is a kernel of some continuous linear automorphism on \(L_p\)). Let \((f_n)\) be a sequence of independent random variables which generate the full \(\sigma\)-algebra of Borel sets. Then \(\langle f_n\rangle ^\infty _{n=1}\) is an admissible kernel. It is also shown that if \(X\) is an admissible kernel, then \(L_p/X\) is strictly transitive.

MSC:

46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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