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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)

MSC:
 46B25 Classical Banach spaces in the general theory 47B07 Linear operators defined by compactness properties
Citations:
Zbl 1080.46007; Zbl 0804.46028