Norm convergence of some power series of operators in $L^{p}$ with applications in ergodic theory

• Autorzy: Christophe Cuny
• Języki publikacji: EN

Abstrakty

EN

Let X be a closed subspace of $L^{p}(μ)$, where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $sup_{n∈ ℤ} ||Uⁿ|| < ∞$. Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like $∑_{n≥1} (Uⁿf)/n^{1-α}$, 0 ≤ α < 1, in terms of $||f + ⋯ + U^{n-1}f||_{p}$, generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie.

Kategorie tematyczne

• 42A45: Multipliers
• 37A30: Ergodic theorems, spectral theory, Markov operators
• 42B25: Maximal functions, Littlewood-Paley theory
• 47B40: Spectral operators, decomposable operators, well-bounded operators, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2010
• Tom: 200
• Numer: 1
• Strony: 1-29

Daty

wydano

2010

Twórcy

autor

Christophe Cuny

• Équipe ERIM, University of New Caledonia, B.P. R4, 98800 Nouméa, New Caledonia

Identyfikatory

DOI

10.4064/sm200-1-1