标题： 立即可微半群的半群增长与预解衰减之间的关系 显示英文标题

标题： Relation between semigroup growth and resolvent decay for immediately differentiable semigroups

摘要： 我们研究当$t \downarrow 0$时$\|AT(t)\|$的增长速率，对于一个立即可微的$C_0$-半群$(T(t))_{t \geq 0}$，其生成元为$A$。 我们假设半群生成元的预解式在虚轴上的衰减速率由正增长函数描述，这使得可以在比多项式更精细的尺度上进行估计。 首先，在巴拿赫空间设置中，我们给出了半群增长的下界和上界。 接下来，我们改进了希尔伯特空间半群的上界估计。 最后，对于希尔伯特空间上的正规算子半群以及 $L^p$-空间上的乘法 $C_0$-半群，我们建立了一个估计，精确地捕捉了 $\|AT(t)\|$随 $t \downarrow 0$的渐近行为。 显示英文摘要

摘要： We study rates of growth of $\|AT(t)\|$ as $t \downarrow 0$ for an immediately differentiable $C_0$-semigroup $(T(t))_{t \geq 0}$ with generator $A$. We assume that the resolvent of the semigroup generator decays on the imaginary axis at rates described by functions of positive increase, which enable estimates on scales finer than polynomial ones. First, in the Banach space setting, we present lower and upper bounds for the semigroup growth. Next, we improve the upper estimate for Hilbert space semigroups. Finally, for semigroups of normal operators on Hilbert spaces and multiplication $C_0$-semigroups on $L^p$-spaces, we establish an estimate that exactly captures the asymptotic behavior of $\|AT(t)\|$ as $t \downarrow 0$.