论文标题
Banach操作员的商代数与非古典近似特性有关的理想
Quotient algebras of Banach operator ideals related to non-classical approximation properties
论文作者
论文摘要
我们研究商代数$ \ mathfrak {a} _x^{\ Mathcal I}：= \ nathcal i（x）/\ overline {\ Mathcal f（x）}^{|| \ cdot || \ cdot || _ {\ cdot || _ {\ Mathcal i}} $ banach Operator $ banach Opertor $ banach Opitator $ \ \ \ \ $ x $是一个失败的Banach空间，该空间失败了$ \ Mathcal I $ -Approximation属性。主要结果涉及nilpotent代数$ \ mathfrak a_x^{\ Mathcal {qn} _p} $和$ \ Mathfrak a_x a_x^{\ Mathcal {sk} _p} _p} __p $ p $ p $ -nuclear Operator $ p $ -compact运算符$ \ Mathcal {SK} _p $。结果包括以下内容：（i）如果$ x $具有cotype 2，则$ \ mathfrak a_x^{\ mathcal {qn} _p} = \ {0 \ {0 \} $ for every $ p \ ge 1 $; （ii）如果$ x^*$具有cotype 2，则每个$ p \ ge 1 $; （iii）$ \ Mathfrak a_x^{\ Mathcal {\ Mathcal {Qn} _p} $和$ \ Mathfrak a_x^{\ Mathcal {sk} _p} $ p \ neq 2 $ p \最大\ max p/p/ \ rceil \} $，其中$ \ left \ lceil p/2 \ right \ rceil $表示\ mathbb n $中的最小$ n \，这样$ n \ ge p/2 $; （iv）对于每$ p> 2 $，都有一个封闭的子空间$ x \ subset C_0 $，这样，$ \ mathfrak a_x^{\ mathcal {qn} _p} $和$ \ mathfrak a_x^{\ mathcal {\ mathcal {sk} _p} $均包含一个数量不计的封闭式封面的理想序列。此外，我们的方法产生了封闭的子空间$ x \ subset C_0 $，使得紧凑型的可与代数$ \ mathfrak a_x = \ mathcal k（x）/\ mathcal a（x）$包含两个无与伦比的无限无限链的无限制封闭封闭的理想。
We investigate the quotient algebra $\mathfrak{A}_X^{\mathcal I}:=\mathcal I(X)/\overline{\mathcal F(X)}^{||\cdot||_{\mathcal I}}$ for Banach operator ideals $\mathcal I$ contained in the ideal of the compact operators, where $X$ is a Banach space that fails the $\mathcal I$-approximation property. The main results concern the nilpotent quotient algebras $\mathfrak A_X^{\mathcal{QN}_p}$ and $\mathfrak A_X^{\mathcal{SK}_p}$ for the quasi $p$-nuclear operators $\mathcal{QN}_p$ and the Sinha-Karn $p$-compact operators $\mathcal{SK}_p$. The results include the following: (i) if $X$ has cotype 2, then $\mathfrak A_X^{\mathcal{QN}_p}=\{0\}$ for every $p\ge 1$; (ii) if $X^*$ has cotype 2, then $\mathfrak A_X^{\mathcal{SK}_p}=\{0\}$ for every $p\ge 1$; (iii) the exact upper bound of the index of nilpotency of $\mathfrak A_X^{\mathcal{QN}_p}$ and $\mathfrak A_X^{\mathcal{SK}_p}$ for $p\neq 2$ is $\max\{2,\left \lceil p/2 \right \rceil\}$, where $\left \lceil p/2 \right \rceil$ denotes the smallest $n\in\mathbb N$ such that $n\ge p/2$; (iv) for every $p>2$ there is a closed subspace $X\subset c_0$ such that both $\mathfrak A_X^{\mathcal{QN}_p}$ and $\mathfrak A_X^{\mathcal{SK}_p}$ contain a countably infinite decreasing chain of closed ideals. In addition, our methods yield a closed subspace $X\subset c_0$ such that the compact-by-approximable algebra $\mathfrak A_X=\mathcal K(X)/\mathcal A(X)$ contains two incomparable countably infinite chains of nilpotent closed ideals.
