Extremal Growth of Multiple Toeplitz Operators and Applications to Numerical Stability of Approximation Schemes

TL;DR

This paper investigates the extremal growth of multiple Toeplitz operators, establishing bounds relating their power growth to resolvent conditions, with applications to the numerical stability of approximation schemes in PDEs.

Contribution

It introduces new bounds linking Toeplitz operator growth to Kreiss-Ritt resolvent conditions using reproducing kernel techniques, applicable to complex operators in numerical analysis.

Findings

Power bounds are bounded by polynomial functions of resolvent conditions.

Explicit reproducing kernel methods effectively analyze non-standard Toeplitz operators.

Results apply to polynomial-symbol Toeplitz operators in numerical PDE solutions.

Abstract

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.