The conversion of resolvent conditions into semigroup estimates is crucial in the stability analysis of hyperbolic partial differential equations. For two families of multiple Toeplitz operators, we relate the power bound with a resolvent condition of Kreiss-Ritt type. Furthermore, we show that the power bound is bounded above by a polynomial of the resolvent condition. The operators under investigation do not fall into a well-understood class, so our analysis utilizes explicit reproducing kernel techniques. Our methods apply \textit{mutatis mutandis} to composites of Toeplitz operators with polynomial symbol, which arise frequently in the numerical solution of initial value problems encountered in science and engineering.

翻译：将预解条件转化为半群估计是双曲型偏微分方程稳定性分析的关键。对于两类多重Toeplitz算子族，我们建立了其幂有界性与Kreiss-Ritt型预解条件之间的关联，并进一步证明了幂有界性可由预解条件的多项式函数界定。由于所研究的算子不属于已充分理解的算子类，我们的分析采用了显式再生核技术。该方法经适当修改后，可应用于以多项式为符号的Toeplitz算子复合结构——这类算子频繁出现在科学与工程领域初值问题数值求解中。