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  • Banach actions preserving unconditional convergence [Elektronski vir]
    Banakh, Taras, 1968- ; Kadec, Vladimir M., 1960-
    Let ▫$A,X,Y$▫ be Banach spaces and ▫$A\times X\to Y$v, ▫$(a,x)\mapsto ax$v, be a continuous bilinear function, called a "Banach action". We say that this action "preserves unconditional convergence" ... if for every bounded sequence ▫$(a_n)_{n\in\omega}$▫ in ▫$A$▫ and unconditionally convergent series ▫$\sum_{n\in\omega}x_n$▫ in ▫$X$▫ the series ▫$\sum_{n\in\omega}a_nx_n$▫ is unconditionally convergent. We prove that a Banach action ▫$A\times X\to Y$▫ preserves unconditional convergence if and only if for any linear functional ▫$y^*\in Y^*$▫ the operator ▫$D_{y^*}:X\to A^*$▫, ▫$D_{y^*}(x)(a)=y^*(ax)$▫, is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from ▫$\ell_1$▫ to ▫$\ell_2$▫, we prove that a Banach action ▫$A\times X\to Y$▫ preserves unconditional convergence if ▫$A$▫ is a Hilbert space possessing an orthonormal basis ▫$(e_n)_{n\in\omega}$▫ such that for every ▫$x\in X$▫ the series ▫$\sum_{n\in\omega}e_nx$▫ is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers ▫$p,q,r\in[1,\infty]$▫ with ▫$\frac1r\le\frac1p+\frac1q$▫, the coordinatewise multplication ▫$\ell_p\times\ell_q\to\ell_r$▫ preserves unconditional convergence if and only if one of the following conditions holds: (i) ▫$p\le 2$▫ and ▫$q\le r$▫, (ii) ▫$2<p<q\le r$▫, (iii) ▫$2<p=q<r$▫, (iv) ▫$r=\infty$▫, (v) ▫$2\le q<p\le r$▫, (vi) ▫$q<2<p$▫ and ▫$\frac1p+\frac1q\ge\frac1r+\frac12$▫.
    Vir: Axioms [Elektronski vir]. - ISSN 2075-1680 (Vol. 11, iss. 1, Jan. 2022, art. 13 (9 str.))
    Vrsta gradiva - e-članek
    Leto - 2022
    Jezik - angleški
    COBISS.SI-ID - 94521603

    Povezava(-e):

    https://doi.org/10.3390/axioms11010013
    DOI

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