Abstract
We prove that if the numerical range of a Hilbert space contraction T is in a certain closed convex set of the unit disk which touches the unit circle only at 1, then ||Tn(I-T)||=O(1/nβ) with β∈[1/2,1). For normal contractions the condition is also necessary. Another sufficient condition for β=12, necessary for T normal, is that the numerical range of T be in a disk (z:|z-δ|≤1-δ} for some δ∈(0, 1). As a consequence of results of Seifert, we obtain that a power-bounded T on a Hilbert space satisfies ||Tn(I-T)||=O(1/nβ) with β∈(0, 1] if and only if sup1<|λ|<2|λ-1|1/β||R(λ, T)||<∞. When T is a contraction on L2 satisfying the numerical range condition, it is shown that Tnf/n1-β converges to 0 a.e. with a maximal inequality, for every f∈L2. An example shows that in general a positive contraction T on L2 may have an f≥0 with limsupTnf/lognn=∞ a.e.
Original language | English |
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Pages (from-to) | 1196-1213 |
Number of pages | 18 |
Journal | Journal of Mathematical Analysis and Applications |
Volume | 436 |
Issue number | 2 |
DOIs | |
State | Published - 15 Apr 2016 |
Keywords
- Contractions on L
- Numerical range
- Quasi-sectorial operators
- Rate of convergence of powers
- Resolvent estimates
- Ritt operators
ASJC Scopus subject areas
- Analysis
- Applied Mathematics