Automatic continuity of operator semigroups in the Calkin algebra
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910385728126976 |
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| author | Kochanek, Tomasz |
| author_facet | Kochanek, Tomasz |
| contents | We study operator semigroups in the Calkin algebra $\mathcal{Q}(\mathcal{H})$, represented as a subalgebra of the algebra of bounded linear operators on a Hilbert space via one of `canonical' Calkin's representations. Using the BDF theory, we associate with any normal $C_0$-semigroup $(q(t))_{t\geq 0}$ in $\mathcal{Q}(\mathcal{H})$ 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 $(q(t))_{t\geq 0}$. Then we show that, in natural circumstances, if $(q(t))_{t\geq 0}$ is continuous in the strong operator topology, then it is actually uniformly continuous, although there are $C_0$-semigroups in $\mathcal{Q}(\mathcal{H})$ that are not uniformly continuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17956 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Automatic continuity of operator semigroups in the Calkin algebra Kochanek, Tomasz Functional Analysis Operator Algebras We study operator semigroups in the Calkin algebra $\mathcal{Q}(\mathcal{H})$, represented as a subalgebra of the algebra of bounded linear operators on a Hilbert space via one of `canonical' Calkin's representations. Using the BDF theory, we associate with any normal $C_0$-semigroup $(q(t))_{t\geq 0}$ in $\mathcal{Q}(\mathcal{H})$ 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 $(q(t))_{t\geq 0}$. Then we show that, in natural circumstances, if $(q(t))_{t\geq 0}$ is continuous in the strong operator topology, then it is actually uniformly continuous, although there are $C_0$-semigroups in $\mathcal{Q}(\mathcal{H})$ that are not uniformly continuous. |
| title | Automatic continuity of operator semigroups in the Calkin algebra |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2403.17956 |