论文标题
紧凑的操作员半群的扰动
Compact perturbations of operator semigroups
论文作者
论文摘要
我们研究了Calkin代数$ \ Mathscr {Q}（\ Mathcal {H}）$的操作员半群的提升问题，我们的方法主要基于Brown--Douglas--fillmore-fillmore理论。 With any normal $C_0$-semigroup $(q(t))_{t\geq 0}$ in $\mathscr{Q}(\mathcal{H})$ we associate an extension $Γ\in\mathrm{Ext}(Δ)$, where $Δ$ is the inverse limit of certain compact metric spaces defined purely in terms of the spectrum $（q（t））的发电机的$σ（a）_ {t \ geq 0} $。通过使用Milnor的确切序列，我们表明，如果每个$ Q（t）$都有正常的升力，那么问题是否会降低$γ$是否会减少相应的首次派生函数是否消失。借助Crisp属性和Kasparov的技术定理，我们提供了$σ（a）$的几何条件，这些条件可以保证$γ$的分裂。如果$δ$是一个完美的紧凑型公制空间，我们以这种方式获得了$ C_0 $ -Semigroup $（q（t））_ {t \ geq 0} $，它可以在Dimatic Primations上提升$（q（q（t））_ {t \ geq 0} $。
We study lifting problems for operator semigroups in the Calkin algebra $\mathscr{Q}(\mathcal{H})$, our approach being mainly based on the Brown--Douglas--Fillmore theory. With any normal $C_0$-semigroup $(q(t))_{t\geq 0}$ in $\mathscr{Q}(\mathcal{H})$ we associate an extension $Γ\in\mathrm{Ext}(Δ)$, where $Δ$ is the inverse limit of certain compact metric spaces defined purely in terms of the spectrum $σ(A)$ of the generator of $(q(t))_{t\geq 0}$. By using Milnor's exact sequence, we show that if each $q(t)$ has a normal lift, then the question whether $Γ$ is trivial reduces to the question whether the corresponding first derived functor vanishes. With the aid of the CRISP property and Kasparov's Technical Theorem, we provide geometric conditions on $σ(A)$ which guarantee splitting of $Γ$. If $Δ$ is a perfect compact metric space, we obtain in this way a $C_0$-semigroup $(Q(t))_{t\geq 0}$ which lifts $(q(t))_{t\geq 0}$ on dyadic rationals.
