论文标题
亚对称碱具有分解特性
Subsymmetric bases have the factorization property
论文作者
论文摘要
我们表明，Banach Space $ X $的每个中型Schauder基础$（e_j）$都有分解属性，即，通过每个有界的操作员$ t \ colon x \ to x $带有$Δ$Δ$ -large-large-large-large-diogonal（就是$ \ iff_jj | \ langle te_j | $（e_j^*）$是$（e_j）$）的生物表达功能。即使$ x $是一个不可分割的双空间，具有弱$^*$ schauder $（e_j）$，我们证明，如果$（e_j）$是non-$ \ ell^1 $ - splicing（没有偏见的$ \ ell^ell^1 $ -semented $ \ ell^1 $ -semence in $ x $ x $），则$（e__j $ __jj）$（e_jj）。对于所有$ 1 \ leq p \ leq \ infty $的$ \ ell^p $ - 单向总和。此外，我们找到了一个无条件基础的条件$（e_j）_ { $δ$-large diagonal can be inverted when restricted to $X_σ= [e_j : j\inσ]$ for a "large" set $σ\subset \{1,\ldots,n\}$ (restricted invertibility of $T$; see Bourgain and Tzafriri [Israel J. Math. 1987, London Math. Soc. Lecture Note Ser. 1989).然后，我们将此结果应用于亚对称基础，以获取在任何空间上定义的$δ$ large对角的$ t $，并以$ x_n $ $ x_n $ $ x_n $ $（e_j）$（e_j）$倒在$x_σ$上，对于$x_σ$，对于一些$σ$，带有$σ$，带有$ quent $ fith $ | fith $ | c | fen | f e | fen | fen | fen | fen $ | fecq c n^^{1/4} $。
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 $T\colon X\to X$ with a $δ$-large diagonal (that is $\inf_j |\langle Te_j, e_j^*\rangle| \geq δ> 0$, 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 $1\leq p\leq \infty$. 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 $T\colon X_n\to X_n$ with $δ$-large diagonal can be inverted when restricted to $X_σ= [e_j : j\inσ]$ for a "large" set $σ\subset \{1,\ldots,n\}$ (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 $δ$-large diagonal defined on any space $X_n$ with a subsymmetric basis $(e_j)$ can be inverted on $X_σ$ for some $σ$ with $|σ|\geq c n^{1/4}$.
