论文标题
$ l（l_p）$中的封闭理想数量
The number of closed ideals in $L(L_p)$
论文作者
论文摘要
我们表明，在Banach代数$ L（l_p（0,1））$，$ 1 <p \ not = 2 <\ infty $中，有$ 2^{2^{\ aleph_0}} $不同的封闭理想。这解决了A. Pietsch 1978年的“操作员理想”中的问题。证明与在Banach空间上有界操作员的空间中产生封闭理想的其他方法完全不同。特别是，理想不包含在严格的奇异运营商中，但不包含对非希尔伯特人的子空间的投影。我们给出了一个无条件基础的空间的标准，即具有$ 2^{2^{\ aleph_0}} $，就单个操作员在该空间中的存在而封闭了理想，并具有一些特殊的渐近性属性。然后，我们表明，对于$ 1 <q <2 $，Rosenthal的空间$ {\ frak x} _q $，这是$ l_q（0,1）$的补充子空间的同构，承认了运营商。
We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1<p\not= 2<\infty$. This solves a problem in A. Pietsch's 1978 book "Operator Ideals". The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non Hilbertian. We give a criterion for a space with an unconditional basis to have $2^{2^{\aleph_0}}$ closed ideals in terms of the existence of a single operator on the space with some special asymptotic properties. We then show that for $1<q<2$ the space ${\frak X}_q$ of Rosenthal, which is isomorphic to a complemented subspace of $L_q(0,1)$, admits such an operator.
