论文标题
几乎不相交的家族引起的Banach空间上的运营商内核
Kernels of operators on Banach spaces induced by almost disjoint families
论文作者
论文摘要
Let~$\mathcal{A}$ be an almost disjoint family of subsets of an infinite set~$Γ$, and denote by~$X_{\mathcal{A}}$ the closed subspace of~$\ell_\infty(Γ)$ spanned by the indicator functions of intersections of finitely many sets in~$\mathcal{A}$.我们表明，如果〜$ \ MATHCAL {a} $具有大于〜$γ$的基数，则封闭的子空间〜$ x _ {\ Mathcal {\ Mathcal {a}} $由表单$ \ bigCap_ {j = 1}}}}^n+1} $ n $ n $ n+1} $ n $ n的指示器函数跨越$ a_1，\ ldots，a_ {n+1} \ in \ mathcal {a} $是不同的，不能是任何有界操作员的内核\ mbox {$ x _ {\ Mathcal {a}}} \ rightArrow \ rightArrow \ rightArrow \ ell el _ {\ infty} {\ infty}（\ infty}（\ infty}（γ）$}}。结果，我们推断出子空间\ [\ bigl \ {x \ in \ ell _ {\ infty}（γ）：\ text {set} \ \ \ {γ\ inγ：\ lerver x（lvert x（γ） \ text {对于每个} \ \ varepsilon> 0 \ bigr \} \]的〜$ \ ell_ \ infty（γ）$不是〜$ \ ell_ \ infty（γ）$上的任何有界操作员的内核；这概括了Kalton和Pełczyński和Sudakov的结果。 对于Banach空间〜$ \ ell_ \ infty^c（γ）$的情况更为复杂，该^c（γ）$的受支持的，有限的功能定义在无数集〜$γ$上。我们表明，在\ textsf {zfc}中是不可决定的，在〜$ \ ell_ \ eld_ \ infty^c（ω_1）$上是否在〜$ c_0（ω_1）上消失的每个有界操作员是否必须在form〜$ c_0（ω_1）上消失。
Let~$\mathcal{A}$ be an almost disjoint family of subsets of an infinite set~$Γ$, and denote by~$X_{\mathcal{A}}$ the closed subspace of~$\ell_\infty(Γ)$ spanned by the indicator functions of intersections of finitely many sets in~$\mathcal{A}$. We show that if~$\mathcal{A}$ has cardinality greater than~$Γ$, then the closed subspace of~$X_{\mathcal{A}}$ spanned by the indicator functions of sets of the form $\bigcap_{j=1}^{n+1}A_j$, where $n\in\N$ and $A_1,\ldots,A_{n+1}\in\mathcal{A}$ are distinct, cannot be the kernel of any bounded operator \mbox{$X_{\mathcal{A}}\rightarrow \ell_{\infty}(Γ)$}. As a consequence, we deduce that the subspace \[ \bigl\{ x\in \ell_{\infty}(Γ) : \text{the set}\ \{γ\in Γ: \lvert x(γ)\rvert > \varepsilon \}\ \text{has cardinality smaller than}\ Γ \text{for every}\ \varepsilon>0\bigr\} \] of~$\ell_\infty(Γ)$ is not the kernel of any bounded operator on~$\ell_\infty(Γ)$; this generalises results of Kalton and of Pełczyński and Sudakov. The situation is more complex for the Banach space~$\ell_\infty^c(Γ)$ of countably supported, bounded functions defined on an uncountable set~$Γ$. We show that it is undecidable in \textsf{ZFC} whether every bounded operator on~$\ell_\infty^c(ω_1)$ which vanishes on~$c_0(ω_1)$ must vanish on a subspace of the form~$\ell_\infty^c(A)$ for some uncountable subset~$A$ of~$ω_1$.
