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Mathematics > Functional Analysis
Title: Compact perturbations of operator semigroups
(Submitted on 10 Mar 2022 (v1), last revised 13 Mar 2023 (this version, v2))
Abstract: 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 $\Gamma\in\mathrm{Ext}(\Delta)$, where $\Delta$ is the inverse limit of certain compact metric spaces defined purely in terms of the spectrum $\sigma(A)$ 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 $\Gamma$ 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 $\sigma(A)$ which guarantee splitting of $\Gamma$. If $\Delta$ 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.
Submission history
From: Tomasz Kochanek [view email][v1] Thu, 10 Mar 2022 20:58:37 GMT (41kb)
[v2] Mon, 13 Mar 2023 18:36:43 GMT (46kb)
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