Duality problem for disjointly homogeneous rearrangement invariant spaces. (English) Zbl 1415.46020
This paper is motivated by a previous one of J. Flores et al. [J. Funct. Anal. 266, No. 9, 5858–5885 (2014; Zbl 1315.46024)].
A Banach lattice \(E\) is disjointly homogeneous (shortly DH) if two arbitrary normalized disjoint sequences from \(E\) contain equivalent subsequences.
For \(1\leq p \leq \infty,\) \(E\) is called \(p\)-disjointly homogeneous (shortly \(p\)-DH) if each normalized disjoint sequence has a subsequence equivalent in \(E\) to the unit vector basis of \(\ell_{p}\) (\(c_{0}\) when \(p=\infty\)).
Considering a question from the above mentioned paper, the author proves as the main result of the present paper:
Theorem 2. For every \(1< p < \infty\), there exists a \(p\)-DH reflexive r.i. space \(X_{p}\) on \([0,1]\) such that its dual space \(X_{p}^{\ast}\) is not DH.
The proof uses interpolation theory, more precisely, the Lions-Peetre interpolation spaces.
Another interesting result is
Theorem 5. Let \(X\) be a reflexive r.i. space on \((0,\infty)\). Then the following conditions are equivalent:
(a) \(X\) and \(X^{\ast}\) are DH;
(b) \(X=L_{p}(0,\infty)\) for some \(1< p< \infty\).
A Banach lattice \(E\) is disjointly homogeneous (shortly DH) if two arbitrary normalized disjoint sequences from \(E\) contain equivalent subsequences.
For \(1\leq p \leq \infty,\) \(E\) is called \(p\)-disjointly homogeneous (shortly \(p\)-DH) if each normalized disjoint sequence has a subsequence equivalent in \(E\) to the unit vector basis of \(\ell_{p}\) (\(c_{0}\) when \(p=\infty\)).
Considering a question from the above mentioned paper, the author proves as the main result of the present paper:
Theorem 2. For every \(1< p < \infty\), there exists a \(p\)-DH reflexive r.i. space \(X_{p}\) on \([0,1]\) such that its dual space \(X_{p}^{\ast}\) is not DH.
The proof uses interpolation theory, more precisely, the Lions-Peetre interpolation spaces.
Another interesting result is
Theorem 5. Let \(X\) be a reflexive r.i. space on \((0,\infty)\). Then the following conditions are equivalent:
(a) \(X\) and \(X^{\ast}\) are DH;
(b) \(X=L_{p}(0,\infty)\) for some \(1< p< \infty\).
Reviewer: Nicolae Popa (Bucureşti)
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46B70 | Interpolation between normed linear spaces |
| 46B42 | Banach lattices |
Keywords:
rearrangement invariant space; disjointly homogeneous lattice; interpolation space; dual spaceCitations:
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