We relate the power bound and a resolvent condition of Kreiss-Ritt type and characterize the extremal growth of two families of products of three Toeplitz operators on the Hardy space that contain infinitely many points in their spectra. Since these operators do not fall into a well-understood class, we analyze them through explicit techniques based on properties of Toeplitz operators and the structure of the Hardy space. Our methods apply mutatis mutandis to operators of the form $T_{g(z)}^{-1}T_{f(z)}T_{g(z)}$ where $f(z)$ is a polynomial in $z$ and $\bar{z}$ and $g(z)$ is a polynomial in $z$. This collection of operators arises in the numerical solution of the Cauchy problem for linear ordinary, partial, and delay differential equations that are frequently used as models for processes in the sciences and engineering. Our results provide a framework for the stability analysis of existing numerical methods for new classes of linear differential equations as well as the development of novel approximation schemes.

翻译：我们将幂有界性与Kreiss-Ritt型预解条件建立联系，并刻画Hardy空间上两个包含谱中无穷多点的三Toeplitz算子乘积族的极值增长。由于这些算子不属于已被充分理解的算子类，我们基于Toeplitz算子的性质与Hardy空间结构，通过显式技巧对其进行分析。我们的方法可类推至形如$T_{g(z)}^{-1}T_{f(z)}T_{g(z)}$的算子，其中$f(z)$是$z$和$\bar{z}$的多项式，$g(z)$是$z$的多项式。这类算子出现在线性常微分方程、偏微分方程及延迟微分方程柯西问题的数值求解中，而这些方程常被用作科学与工程领域中过程模型的数学描述。我们的结果为现有数值方法针对新型线性微分方程的稳定性分析，以及新型逼近方案的发展提供了理论框架。