论文标题
L_P收入和类似操作员拓扑的通用属性
Generic properties of l_p-contractions and similar operator topologies
论文作者
论文摘要
如果$ x $是可分离的反射巴纳克空间，那么$ \ Mathcal {b}（x）$上有几种天然波兰拓扑，这是$ x $上的一组收缩操作员（没有一个显然是“比其他人更自然”），并且在Baire contractions -contraction contractions contractions contractions contractions contraction contractions contractions contractions contractions contractions。因此，调查一般属性（即稳定集）在$ \ Mathcal {B}（x）$上所选择的拓扑的程度，即合并设置，这是有道理的。在本文中，我们专注于$ \ ell_p \，$ - $ \，$ spaces，$ 1 <p \ neq 2 <\ infty $。我们表明，对于$ \ Mathcal B_1（\ ell_p）$上的一些天然波兰拓扑，相当的套件实际上是相同的；我们的主要结果表明，对于$ p = 3 $或$ 3/2 $，以及在实际情况下，所有拓扑均以$ \ Mathcal b_1（\ ell_p）$为单位，位于弱操作员拓扑与强$^*$ operator拓扑之间共享相同的comeager套件。我们的研究依赖于$ \ Mathcal {B} _1（\ ell_p）$上两个不同拓扑的身份图的连续点的考虑。我们主要结果证明的另一个基本要素是对特殊类型的有限维度收缩的规范向量进行仔细检查。
If $X$ is a separable reflexive Banach space, there are several natural Polish topologies on $\mathcal{B}(X)$, the set of contraction operators on $X$ (none of which being clearly ``more natural'' than the others), and hence several a priori different notions of genericity -- in the Baire category sense -- for properties of contraction operators. So it makes sense to investigate to which extent the generic properties, i.e. the comeager sets, really depend on the chosen topology on $\mathcal{B}(X)$. In this paper, we focus on $\ell_p\,$-$\,$spaces, $1<p\neq 2<\infty$. We show that for some pairs of natural Polish topologies on $\mathcal B_1(\ell_p)$, the comeager sets are in fact the same; and our main result asserts that for $p=3$ or $3/2$ and in the real case, all topologies on $\mathcal B_1(\ell_p)$ lying between the Weak Operator Topology and the Strong$^*$ Operator Topology share the same comeager sets. Our study relies on the consideration of continuity points of the identity map for two different topologies on $\mathcal{B}_1 (\ell_p)$. The other essential ingredient in the proof of our main result is a careful examination of norming vectors for finite-dimensional contractions of a special type.
