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

This paper pioneers the application of reproducing kernel spaces in proving the stability of numerical methods for approximating linear differential equations. We analyze families of products of Hardy space Toeplitz operators that do not fall into a well-understood class of operators through explicit techniques, many of which are based upon exploiting the reproducing kernel Hilbert space structure of the Hardy space on the unit disk and properties of Toeplitz operators. This results in bounds that would not be feasible through more general functional analytic techniques. In particular, for these operators, we relate the power bound and a resolvent condition of Kreiss-Ritt type since it is well-known that such a relationship implies the stability of certain methods for the numerical solution of linear differential equations. 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 $\overline{z}$ and $g(z)$ is a polynomial in $z$, which arise frequently in the numerical solution of the Cauchy problem for linear ordinary, partial, and delay differential equations used as models for processes in science and engineering.