Interpolation and non-dilatable families of $$\mathcal {C}_{0}$$ -semigroups

IF 1.1 2区 数学 Q1 MATHEMATICS
Raj Dahya
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引用次数: 0

Abstract

We generalise a technique of Bhat and Skeide (J Funct Anal 269:1539–1562, 2015) to interpolate commuting families \(\{S_{i}\}_{i \in \mathcal {I}}\) of contractions on a Hilbert space \(\mathcal {H}\), to commuting families \(\{T_{i}\}_{i \in \mathcal {I}}\) of contractive \(\mathcal {C}_{0}\)-semigroups on \(L^{2}(\prod _{i \in \mathcal {I}}\mathbb {T}) \otimes \mathcal {H}\). As an excursus, we provide applications of the interpolations to time-discretisation and the embedding problem. Applied to Parrott’s construction (1970), we then demonstrate for \(d \in \mathbb {N}\) with \(d \ge 3\) the existence of commuting families \(\{T_{i}\}_{i=1}^{d}\) of contractive \(\mathcal {C}_{0}\)-semigroups which admit no simultaneous unitary dilation. As an application of these counter-examples, we obtain the residuality wrt.the topology of uniform \(\textsc {wot}\)-convergence on compact subsets of \(\mathbb {R}_{\ge 0}^{d}\) of non-unitarily dilatable and non-unitarily approximable d-parameter contractive \(\mathcal {C}_{0}\)-semigroups on separable infinite-dimensional Hilbert spaces for each \(d \ge 3\). Similar results are also developed for d-tuples of commuting contractions. And by building on the counter-examples of Varopoulos-Kaijser (1973–74), a 0-1-result is obtained for the von Neumann inequality. Finally, we discuss applications to rigidity as well as the embedding problem, viz. that ‘typical’ pairs of commuting operators can be simultaneously embedded into commuting pairs of \(\mathcal {C}_{0}\)-semigroups, which extends results of Eisner (2009–2010).

Abstract Image

$$\mathcal {C}_{0}$ -semigroups 的插值和不可稀疏族
我们将巴特和斯基德(J Funct Anal 269:1539-1562, 2015)来插值希尔伯特空间 \(\mathcal {H}\) 上收缩的换向族 \(\{S_{i}\}_{i \in \mathcal {I}}\)、到 \(L^{2}(\prod _{i \in \mathcal {I}\mathbb {T}) \otimes \mathcal {H}/)上的收缩(\mathcal {C}_{0}/)-半群的共摂族 \(\{T_{i}\}_{i \in \mathcal {I}\mathbb {T}).作为一个小插曲,我们将插值应用于时间离散化和嵌入问题。应用于帕洛特的构造(1970),我们证明了对于具有(d ge 3)的(d \in \mathbb {N})收缩(\mathcal {C}_{0})-半群,存在不允许同时进行单元扩张的共相族(\{T_{i}\}_{i=1}^{d}\)。作为这些反例的一个应用,我们得到了关于均匀 \(\mathcal {C}_{0}\) 的拓扑的剩余性。的紧凑子集上的均匀收敛拓扑。类似的结果也适用于换向收缩的 d 元组。通过建立在 Varopoulos-Kaijser (1973-74) 反例的基础上,我们得到了 von Neumann 不等式的 0-1 结果。最后,我们讨论了刚性以及嵌入问题的应用,即 "典型的 "换向算子对可以同时嵌入到换向对(\mathcal {C}_{0}\)-半群中,这扩展了艾斯纳(2009-2010)的结果。
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来源期刊
CiteScore
2.00
自引率
8.30%
发文量
67
审稿时长
>12 weeks
期刊介绍: The Banach Journal of Mathematical Analysis (Banach J. Math. Anal.) is published by Birkhäuser on behalf of the Tusi Mathematical Research Group. Banach J. Math. Anal. is a peer-reviewed electronic journal publishing papers of high standards with deep results, new ideas, profound impact, and significant implications in all areas of functional analysis and operator theory and all modern related topics. Banach J. Math. Anal. normally publishes survey articles and original research papers numbering 15 pages or more in the journal’s style. Shorter papers may be submitted to the Annals of Functional Analysis or Advances in Operator Theory.
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