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Register a new userThe factorization property of $ell^infty(X_k)$
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In this paper we consider the following problem: Let $X_k$, be a Banach space with a normalized basis $(e_{(k,j)})_j$, whose biorthogonals are denoted by $(e_{(k,j)}^*)_j$, for $kinmathbb{N}$, let $Z=ell^infty(X_k:kinmathbb{N})$ be their $ell^infty$-sum, and let $T:Zto Z$ be a bounded linear operator, with a large diagonal, i.e. $$inf_{k,j} big|e^*_{(k,j)}(T(e_{(k,j)})big|>0.$$ Under which condition does the identity on $Z$ factor through $T$? The purpose of this paper is to formulate general conditions for which the answer is positive.
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We show that the non-separable Banach space $SL^infty$ is primary. This is achieved by directly solving the infinite dimensional factorization problem in $SL^infty$. In particular, we bypass Bourgains localization method.
The unique maximal ideal in the Banach algebra $L(E)$, $E = (oplus ell^infty(n))_{ell^1}$, is identified. The proof relies on techniques developed by Laustsen, Loy and Read and a dichotomy result for operators mapping into $L^1$ due to Laustsen, Odell, Schlumprecht and Zs{a}k.
We show that every subsymmetric Schauder basis $(e_j)$ of a Banach space $X$ has the factorization property, i.e. $I_X$ factors through every bounded operator $Tcolon Xto X$ with a $delta$-large diagonal (that is $inf_j |langle Te_j, e_j^*rangle| geq delta > 0$, where the $(e_j^*)$ are the biorthogonal functionals to $(e_j)$). Even if $X$ is a non-separable dual space with a subsymmetric weak$^*$ Schauder basis $(e_j)$, we prove that if $(e_j)$ is non-$ell^1$-splicing (there is no disjointly supported $ell^1$-sequence in $X$), then $(e_j)$ has the factorization property. The same is true for $ell^p$-direct sums of such Banach spaces for all $1leq pleq infty$. Moreover, we find a condition for an unconditional basis $(e_j)_{j=1}^n$ of a Banach space $X_n$ in terms of the quantities $|e_1+ldots+e_n|$ and $|e_1^*+ldots+e_n^*|$ under which an operator $Tcolon X_nto X_n$ with $delta$-large diagonal can be inverted when restricted to $X_sigma = [e_j : jinsigma]$ for a large set $sigmasubset {1,ldots,n}$ (restricted invertibility of $T$; see Bourgain and Tzafriri [Israel J. Math. 1987, London Math. Soc. Lecture Note Ser. 1989). We then apply this result to subsymmetric bases to obtain that operators $T$ with a $delta$-large diagonal defined on any space $X_n$ with a subsymmetric basis $(e_j)$ can be inverted on $X_sigma$ for some $sigma$ with $|sigma|geq c n^{1/4}$.
We introduce the concept of strategically reproducible bases in Banach spaces and show that operators which have large diagonal with respect to strategically reproducible bases are factors of the identity. We give several examples of classical Banach spaces in which the Haar system is strategically reproducible: multi-parameter Lebesgue spaces, mixed-norm Hardy spaces and most significantly the space $L^1$. Moreover, we show the strategical reproducibility is inherited by unconditional sums.
Let $D$ be an integral domain. A nonzero nonunit $a$ of $D$ is called a valuation element if there is a valuation overring $V$ of $D$ such that $aVcap D=aD$. We say that $D$ is a valuation factorization domain (VFD) if each nonzero nonunit of $D$ can be written as a finite product of valuation elements. In this paper, we study some ring-theoretic properties of VFDs. Among other things, we show that (i) a VFD $D$ is Schreier, and hence ${rm Cl}_t(D)={0}$, (ii) if $D$ is a P$v$MD, then $D$ is a VFD if and only if $D$ is a weakly Matlis GCD-domain, if and only if $D[X]$, the polynomial ring over $D$, is a VFD and (iii) a VFD $D$ is a weakly factorial GCD-domain if and only if $D$ is archimedean. We also study a unique factorization property of VFDs.
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