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Almost everywhere convergence of generalized ergodic transforms for invertible power-bounded operators in L p

Christophe Cuny (2011)

Colloquium Mathematicae

We show that some results of Gaposhkin about a.e. convergence of series associated to a unitary operator U acting on L²(X,Σ,μ) (μ is a σ-finite measure) may be extended to the case where U is an invertible power-bounded operator acting on L p ( X , Σ , μ ) , p > 1. The proofs make use of the spectral integration initiated by Berkson-Gillespie and, more specifically, of recent results of the author.

An M q ( ) -functional calculus for power-bounded operators on certain UMD spaces

Earl Berkson, T. A. Gillespie (2005)

Studia Mathematica

For 1 ≤ q < ∞, let q ( ) denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes q ( ) as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction q >...

Automatic extensions of functional calculi

Ralph deLaubenfels (1995)

Studia Mathematica

Given a Banach algebra ℱ of complex-valued functions and a closed, linear (possibly unbounded) densely defined operator A, on a Banach space, with an ℱ functional calculus we present two ways of extending this functional calculus to a much larger class of functions with little or no growth conditions. We apply this to spectral operators of scalar type, generators of bounded strongly continuous groups and operators whose resolvent set contains a half-line. For f in this larger class, one construction...

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