Compact perturbations of operator semigroups
Tomasz Kochanek

TL;DR
This paper investigates lifting problems for operator semigroups in the Calkin algebra using K-theoretic tools, providing conditions under which such lifts exist and constructing specific lifted semigroups.
Contribution
It introduces a new approach to lifting operator semigroups in the Calkin algebra via extension theory and spectral analysis, with geometric conditions for splitting extensions.
Findings
Extension $ ext{Gamma}$ relates to the spectrum of the generator.
Splitting of $ ext{Gamma}$ depends on the vanishing of a derived functor.
Constructs a lifted $C_0$-semigroup on dyadic rationals.
Abstract
We study lifting problems for operator semigroups in the Calkin algebra , our approach being mainly based on the Brown--Douglas--Fillmore theory. With any normal -semigroup in we associate an extension , where is the inverse limit of certain compact metric spaces defined purely in terms of the spectrum of the generator of . By using Milnor's exact sequence, we show that if each has a normal lift, then the question whether 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 which guarantee splitting of . If is a perfect compact metric space, we obtain in…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMathematical Dynamics and Fractals · Advanced Topics in Algebra · Quantum chaos and dynamical systems
