Disjointly homogeneous rearrangement invariant spaces via interpolation. (English) Zbl 1328.46022
Summary: A Banach lattice \(E\) is called \(p\)-disjointly homogeneous, \(1\leq p\leq\infty\), when every sequence of pairwise disjoint normalized elements in \(E\) has a subsequence equivalent to the unit vector basis of \(\ell_p\). Employing methods from interpolation theory, we clarify which r.i. spaces on \([0,1]\) are \(p\)-disjointly homogeneous. In particular, for every \(1<p<\infty\) and any increasing concave function \(\varphi\) on \([0,1]\), which is not equivalent to neither 1 nor \(t\), there exists a \(p\)-disjointly homogeneous r.i. space with the fundamental function \(\varphi\). Moreover, it is shown that given \(1<p<\infty\) and an increasing concave function \(\varphi\) with non-trivial dilation indices, there is a unique \(p\)-disjointly homogeneous space among all interpolation spaces between the Lorentz and Marcinkiewicz spaces associated with \(\varphi\).
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46B42 | Banach lattices |
| 46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |
| 46B70 | Interpolation between normed linear spaces |
Keywords:
rearrangement invariant space; Lorentz space; Marcinkiewicz space; Orlicz space; real method of interpolation; Calderón-Lozanovskii spaceReferences:
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