论文标题
通勤$ C_ {0} $ - 具有有限发电机和von Neumann多项式不平等的膨胀
Dilations of commuting $C_{0}$-semigroups with bounded generators and the von Neumann polynomial inequality
论文作者
论文摘要
考虑$ d $通勤$ c_ {0} $ - semigroups（或等效：$ d $ - 参数$ c_ {0} $ - semigroups）在希尔伯特空间上，用于$ d \ in \ mathbb {n} $。在文献（\ textit {cf。} [29，26，27，23，18，25]）中，提供了条件，以分类统一和常规统一扩张的存在。其中一些条件需要检查半群的值，有些仅提供足够的条件，而另一些则涉及验证发电机的复杂属性。通过专注于具有有界发电机的半群，我们在发电机上建立了一个简单自然的条件，\ textit {viz。} \ emph {完整的消散性}自然地扩展了发电机消散性的基本概念。使用非换通向半群的示例，可以证明该特性比消散性更强。作为第一个主要结果，我们证明了完全的耗散性完全表征了常规统一扩张的存在，并将其扩展到任意上下班$ c_ {0} $ - 半群的情况下。我们此外表明，所有多参数$ c_ {0} $ - 半群（带有有界的发电机）都接受了较弱的常规单一扩张概念，并提供了简单的足够的规范标准以完全消散。本文以对von Neumann多项式不平等问题的申请结束，我们为半群设置制定，并对所有$ d \ geq 2 $负面求解。
Consider $d$ commuting $C_{0}$-semigroups (or equivalently: $d$-parameter $C_{0}$-semigroups) over a Hilbert space for $d \in \mathbb{N}$. In the literature (\textit{cf.} [29, 26, 27, 23, 18, 25]), conditions are provided to classify the existence of unitary and regular unitary dilations. Some of these conditions require inspecting values of the semigroups, some provide only sufficient conditions, and others involve verifying sophisticated properties of the generators. By focussing on semigroups with bounded generators, we establish a simple and natural condition on the generators, \textit{viz.} \emph{complete dissipativity}, which naturally extends the basic notion of the dissipativity of the generators. Using examples of non-doubly commuting semigroups, this property can be shown to be strictly stronger than dissipativity. As the first main result, we demonstrate that complete dissipativity completely characterises the existence of regular unitary dilations, and extend this to the case of arbitrarily many commuting $C_{0}$-semigroups. We furthermore show that all multi-parameter $C_{0}$-semigroups (with bounded generators) admit a weaker notion of regular unitary dilations, and provide simple sufficient norm criteria for complete dissipativity. The paper concludes with an application to the von Neumann polynomial inequality problem, which we formulate for the semigroup setting and solve negatively for all $d \geq 2$.
