Extremal Growth of Multiple Toeplitz Operators and Implications for Numerical Stability of Approximation Schemes

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.