Extremal Growth of Powers of Operators Satisfying Resolvent Conditions of Kreiss-Ritt Type
✍ Scribed by Omar El-Fallah; Thomas Ransford
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 172 KB
- Volume
- 196
- Category
- Article
- ISSN
- 0022-1236
No coin nor oath required. For personal study only.
✦ Synopsis
Let E be a compact subset of the unit circle. We determine the extremal rate of growth of ðjjT n jjÞ n51 for Banach-space operators T satisfying the resolvent condition jjðT À lIÞ À1 jj4 const: distðl; EÞ ðjlj > 1Þ:
This includes, as extreme cases, the Kreiss condition E ¼ T and the Ritt condition E ¼ f1g: For intermediate sets E; the cardinality, the measure and the Hausdorff dimension of E all play a roˆle in determining the growth of jjT n jj: As a by-product, we also obtain lower bounds for the Taylor coefficients of functions f holomorphic on the unit disk and satisfying j f ðzÞj5 1 distðz; EÞ ðjzjo1Þ: