AI 中文总结
该研究针对希尔伯特空间上的Kreiss有界C₀半群，改进了其增长估计，证明存在线性增长下的多项式间隙，且不存在通用正指数。
AI 中文摘要
我们证明，希尔伯特空间上的每个Kreiss有界C₀半群(Tₜ)ₜ≥0满足‖Tₜ‖≤C(1+t)^(1−ε_K)（t≥0），其中ε_K>0仅显式依赖于Kreiss常数。这改进了先前已知的O(t/√log(t+1))估计，表明每个Kreiss有界C₀半群都存在线性增长下的真正多项式间隙。结合Eisner和Zwart给出的增长任意接近线性的例子，可知对希尔伯特空间上全体Kreiss有界C₀半群，不存在通用的正指数。
英文摘要
We prove that every Kreiss bounded $C_0$-semigroup $(T_t)_{t\ge0}$ on a Hilbert space satisfies \[ \|T_t\|\le C(1+t)^{1-\varepsilon_K}, \qquad t\ge0, \] where $\varepsilon_K>0$ depends explicitly only on the Kreiss constant. The same conclusion is obtained for positive Kreiss bounded $C_0$-semigroups on $L^p$-spaces, $1<p<\infty$, and in discrete time for Kreiss bounded operators on Hilbert spaces and positive Kreiss bounded operators on $L^p$-spaces. We further obtain a non-quantitative polynomial gap for individually eventually positive Kreiss bounded $C_0$-semigroups on $L^p$-spaces. Finally we prove that every Kreiss bounded operator on a UMD Banach space has a polynomial gap below linear growth
CommentsExpanded version: Added a new section on individually eventually positive Kreiss bounded C0-semigroups on Lp-spaces, establishing a non-quantitative polynomial gap below linear growth. Added a new section proving a polynomial growth gap for Kreiss bounded operators on UMD spaces. 21 pages
