论文标题
P-Schatten预测换向器
p-Schatten commutators of projections
论文作者
论文摘要
令$ h = h _+\ oplus h _- $是在两个无限维度子空间中的复杂希尔伯特空间$ h $的固定正交分解。我们研究了Banach代数$$ {\ cal a}^p = \ {a \ in B（h）：[a，e _+] \ in B_p（h）\}，$ e _+$是$ e _+y $ h _+$ h _ _+$ b的， $ P $ - 可亮的运营商（$ 1 \ le P <\ infty $）。 $ {\ cal a}^p $中的规范是根据上述分解给出的操作员的矩阵条目的规范来定义的。 Space $ p^p $显示为$ {\ cal a}^p $的$ c^\ infty $ submanifold，并且是$ {\ cal a}^p $的统一操作员组的均匀空间。 $ p^p $的连接组件通过九类中的$ p^p $分区，四个离散类和五个基本类别进行了特征： - 前两个对应于有限的等级或共同级别，其连接的组件由论文等级参数为参数； - 接下来的两个离散类带有弗雷德·索（Fredholm）索引，该索引参数为其组件； - 剩下的基本类别，这些类别连接。
Let $H=H_+\oplus H_-$ be a fixed orthogonal decomposition of the complex Hilbert space $H$ in two infinite dimensional subspaces. We study the geometry of the set $P^p$ of selfadjoint projections in the Banach algebra $$ {\cal A}^p=\{A\in B(H): [A,E_+]\in B_p(H)\}, $$ where $E_+$ is the projection onto $H_+$ and $B_p(H)$ is the Schatten ideal of $p$-summable operators ($1\le p <\infty$). The norm in ${\cal A}^p$ is defined in terms of the norms of the matrix entries of the operators given by the above decomposition. The space $P^p$ is shown to be a differentiable $C^\infty$ submanifold of ${\cal A}^p$, and a homogeneous space of the group of unitary operators in ${\cal A}^p$. The connected components of $P^p$ are characterized, by means of a partition of $P^p$ in nine classes, four discrete classes and five essential classes: - the first two corresponding to finite rank or co-rank, with the connected components parametrized by theses ranks; - the next two discrete classes carrying a Fredholm index, which parametrizes its components; - the remaining essential classes, which are connected.
