We extend and deepen the theory of functional calculus for semigroup generators, based on the algebra $\mathcal{B}$ of analytic Besov functions, which we initiated in a previous paper. In particular, we show that our construction of the calculus is optimal in several natural senses. Moreover, we clarify the structure of $\mathcal{B}$ and identify several important subspaces in practical terms. This leads to new spectral mapping theorems for operator semigroups and to wide generalisations of a number of basic results from semigroup theory.

中文翻译：

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Besov函数演算理论：半群的发展和应用

我们基于代数扩展和深化了半群生成器的函数演算理论 $\mathcal{乙}$Besov函数的解析，我们在之前的文章中对此进行了介绍。尤其是，我们证明了微积分的构造在几种自然意义上是最优的。此外，我们阐明了$\mathcal{乙}$并从实际角度确定几个重要的子空间。这导致了针对算子半群的新谱映射定理，并导致了半群理论对许多基本结果的广泛推广。