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More examples of hereditarily \(\ell_p\) Banach spaces. (English) Zbl 1166.46304
Ukr. Mat. Visn. 2, No. 1, 92-108 (2005) and in Ukr. Math. Bull. 2, No. 1, 95-111 (2005).
Recall that a Banach space \(X\) is said to be hereditarily \(\ell_p\) if each infinite-dimensional closed subspace \(Y\) of \(X\) contains a subspace \(Z\subset Y\) which is isomorphic to \(\ell_p\).
Let \(\mathcal{P}\) be a decreasing sequence of reals \(p_1>p_2>\dots>1\). The author constructs a class of hereditarily \(\ell_p\) spaces \(Z_p(\mathcal{P})\). The space \(Z_p(\mathcal{P})\) is a subspace of the \(\ell_p\)-direct sum of the spaces \(\ell_{p_n}\), it has a 1-symmetric basis, and it is not isomorphic to \(\ell_p\). The construction develops further a similar construction by the author in [Proc. Am. Math. Soc. 133, No. 7, 2023–2028 (2005; Zbl 1080.46007)].
Recall that the classical Pitt theorem asserts that, for \(p>q\), all bounded linear operators from \(\ell_p\) to \(\ell_q\) are compact. The author shows that, for \(p>q\), there exist non-compact bounded linear operators from \(\ell_p\) to \(Z_q(\mathcal{P})\) whenever \(\inf p_n\geq p\).
This is a new example of the phenomenon that Pitt’s theorem does not extend to hereditarily \(\ell_p\) spaces.
Reviewer’s remark. The first example of this phenomenon seems to be due to the reviewer [Eesti Tead. Akad. Toim., Füüs. Mat. 40, No. 1, 31–36 (1991; Zbl 0804.46028)]: for \(p>q\), there exist non-compact bounded linear operators from \(\ell_p\) to the Lorentz sequence space \(d(w,q)\) whenever \(w=(w_n)\in\ell_{p/(p-q)}\).
Reviewer: Eve Oja (Tartu)

46B25 Classical Banach spaces in the general theory
47B07 Linear operators defined by compactness properties