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An $M_{q}(𝕋)$-functional calculus for power-bounded operators on certain UMD spaces

100%

Berkson E., Gillespie T.

Studia Mathematica

2005

tom 167

nr 3

245-257

For 1 ≤ q < ∞, let $𝔐_{q}(𝕋)$ 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 $𝔐_{q}(𝕋)$ 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 > 1. Moreover, when taken in conjunction with vector-valued transference, this $𝔐_{q}(𝕋)$-multiplier result shows that if X ∈ ℐ, and U is an invertible power-bounded operator on X, then U has an $𝔐_{q}(𝕋)$-functional calculus for an appropriate range of values of q > 1. The class ℐ includes, in particular, all closed subspaces of the von Neumann-Schatten p-classes $𝓒_{p}$ (1 < p < ∞), as well as all closed subspaces of any UMD lattice of functions on a σ-finite measure space. The $𝔐_{q}(𝕋)$-functional calculus result for ℐ, when specialized to the setting of closed subspaces of $L^{p}(μ)$ (μ an arbitrary measure, 1 < p < ∞), recovers a previous result of the authors.

Spectral decompositions, ergodic averages, and the Hilbert transform

100%

Berkson E., Gillespie T.

Studia Mathematica

2001

tom 144

nr 1

39-61

Let U be a trigonometrically well-bounded operator on a Banach space 𝔛, and denote by ${𝔄ₙ(U)}_{n=1}^{∞}$ the sequence of (C,2) weighted discrete ergodic averages of U, that is, $𝔄ₙ(U) = 1/n ∑_{0<|k|≤n} (1 - |k|/(n+1)) U^{k}$. We show that this sequence ${𝔄ₙ(U)}_{n=1}^{∞}$ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is {x ∈ 𝔛: Ux = x}, and whose null space is the closure of (I - U)𝔛. This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral theory and ergodic operator theory. We also develop a characterization of trigonometrically well-bounded operators by their ability to "transfer" the discrete Hilbert transform to the Banach space setting via (C,1) weighting of Hilbert averages, and these results together with those on weighted ergodic averages furnish an explicit expression for the spectral decomposition of a trigonometrically well-bounded operator U on a Banach space in terms of strong limits of appropriate averages of the powers of U. We also treat the special circumstances where corresponding results can be obtained with the (C,1) and (C,2) weights removed.

Ograniczanie wyników

2 Studia Mathematica

2 Berkson E.

2 Gillespie T.

1 2005

1 2001

Wyniki wyszukiwania

An $M_{q}(𝕋)$-functional calculus for power-bounded operators on certain UMD spaces

Spectral decompositions, ergodic averages, and the Hilbert transform