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Search: MSC category 46B45 ( Banach sequence spaces [See also 46A45] )

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1. CJM Online first

Astashkin, Sergey V.; Lesnik, Karol; Maligranda, Lech
Isomorphic structure of Cesàro and Tandori spaces
We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space $Ces_{\infty}$ and its sequence counterpart $ces_{\infty}$ are isomorphic, which answers the question posted previously. This is rather surprising since $Ces_{\infty}$ (like the known Talagrand's example) has no natural lattice predual. We prove that $ces_{\infty}$ is not isomorphic to ${\ell}_{\infty}$ nor is $Ces_{\infty}$ isomorphic to the Tandori space $\widetilde{L_1}$ with the norm $\|f\|_{\widetilde{L_1}}= \|\widetilde{f}\|_{L_1},$ where $\widetilde{f}(t):= \operatorname{esssup}_{s \geq t} |f(s)|.$ Our investigation involves also an examination of the Schur and Dunford-Pettis properties of Cesàro and Tandori spaces. In particular, using results of Bourgain we show that a wide class of Cesàro-Marcinkiewicz and Cesàro-Lorentz spaces have the latter property.

Keywords:Cesàro and Tandori sequence spaces, Cesàro and Tandori function spaces, Cesàro operator, Banach ideal space, symmetric space, Schur property, Dunford-Pettis property, isomorphism
Categories:46E30, 46B20, 46B42, 46B45

2. CJM 2007 (vol 59 pp. 614)

Labuschagne, C. C. A.
Preduals and Nuclear Operators Associated with Bounded, $p$-Convex, $p$-Concave and Positive $p$-Summing Operators
We use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of $p$-convex, $p$-concave and positive $p$-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

Keywords:$p$-convex operator, $p$-concave operator, $p$-summing operator, Banach space, Banach lattice, nuclear operator, sequence space
Categories:46B28, 47B10, 46B42, 46B45

3. CJM 1999 (vol 51 pp. 309)

Leung, Denny H.; Tang, Wee-Kee
Symmetric sequence subspaces of $C(\alpha)$, II
If $\alpha$ is an ordinal, then the space of all ordinals less than or equal to $\alpha$ is a compact Hausdorff space when endowed with the order topology. Let $C(\alpha)$ be the space of all continuous real-valued functions defined on the ordinal interval $[0, \alpha]$. We characterize the symmetric sequence spaces which embed into $C(\alpha)$ for some countable ordinal $\alpha$. A hierarchy $(E_\alpha)$ of symmetric sequence spaces is constructed so that, for each countable ordinal $\alpha$, $E_\alpha$ embeds into $C(\omega^{\omega^\alpha})$, but does not embed into $C(\omega^{\omega^\beta})$ for any $\beta < \alpha$.

Categories:03E13, 03E15, 46B03, 46B45, 46E15, 54G12

4. CJM 1997 (vol 49 pp. 1242)

Randrianantoanina, Beata
$1$-complemented subspaces of spaces with $1$-unconditional bases
We prove that if $X$ is a complex strictly monotone sequence space with $1$-un\-con\-di\-tion\-al basis, $Y \subseteq X$ has no bands isometric to $\ell_2^2$ and $Y$ is the range of norm-one projection from $X$, then $Y$ is a closed linear span a family of mutually disjoint vectors in $X$. We completely characterize $1$-complemented subspaces and norm-one projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q < \infty$. Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are $1$-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space $X$ is not isomorphic to $\ell_p$ for some $1 \leq p < \infty$ then the only subspaces of $X$ which are $1$-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients.

Categories:46B20, 46B45, 41A65

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