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1
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Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series

100%
Berkson E.
Studia Mathematica
|
2014
|
tom 222
|
nr 2
123-155
EN
Let $f ∈ V_{r}(𝕋) ∪ 𝔐_{r}(𝕋)$, where, for 1 ≤ r < ∞, $V_{r}(𝕋)$ (resp., $𝔐_{r}(𝕋)$) denotes the class of functions (resp., bounded functions) g: 𝕋 → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on 𝕋 (resp., on the dyadic arcs of 𝕋). In the author's recent article [New York J. Math. 17 (2011)] it was shown that if 𝔛 is a super-reflexive space, and E(·): ℝ → 𝔅(𝔛) is the spectral decomposition of a trigonometrically well-bounded operator U ∈ 𝔅(𝔛), then over a suitable non-void open interval of r-values, the condition $f ∈ V_{r}(𝕋)$ implies that the Fourier series $∑_{k=-∞}^{∞} f̂(k)z^{k}U^{k}$ (z ∈ 𝕋) of the operator ergodic "Stieltjes convolution" $𝔖_{U}: 𝕋 → 𝔅(𝔛)$ expressed by $∫t_{[0,2π]}^{⊕} f(ze^{it})dE(t)$ converges at each z ∈ 𝕋 with respect to the strong operator topology. The present article extends the scope of this result by treating the Fourier series expansions of operator ergodic Stieltjes convolutions when, for a suitable interval of r-values, f is a continuous function that is merely assumed to lie in the broader (but less tractable) class $𝔐_{r}(𝕋)$. Since it is known that there are a trigonometrically well-bounded operator U₀ acting on the Hilbert sequence space 𝔛 = ℓ²(ℕ) and a function f₀ ∈ 𝔐₁(𝕋) which cannot be integrated against the spectral decomposition of U₀, the present treatment of Fourier series expansions for operator ergodic convolutions is confined to a special class of trigonometrically well-bounded operators (specifically, the class of disjoint, modulus mean-bounded operators acting on $L^{p}(μ)$, where μ is an arbitrary sigma-finite measure, and 1 < p < ∞). The above-sketched results for operator-valued Stieltjes convolutions can be viewed as a single-operator transference machinery that is free from the power-boundedness requirements of traditional transference, and endows modern spectral theory and operator ergodic theory with the tools of Fourier analysis in the tradition of Hardy-Littlewood, J. Marcinkiewicz, N. Wiener, the (W. H., G. C., and L. C.) Young dynasty, and others. In particular, the results show the behind-the-scenes benefits of the operator ergodic Hilbert transform and its dual conjugates, and encompass the Fourier multiplier actions of $𝔐_{r}(𝕋)$-functions in the setting of $A_{p}$-weighted sequence spaces.
2
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Spectral theory and operator ergodic theory on super-reflexive Banach spaces

100%
Berkson E.
Studia Mathematica
|
2010
|
tom 200
|
nr 3
221-246
EN
On reflexive spaces trigonometrically well-bounded operators have an operator-ergodic-theory characterization as the invertible operators U such that $sup_{n∈ ℕ, z∈ 𝕋} || ∑_{0<|k|≤n} (1 - |k|/(n+1))k^{-1}z^{k}U^{k}|| < ∞$. (*) Trigonometrically well-bounded operators permeate many settings of modern analysis, and this note highlights the advances in both their spectral theory and operator ergodic theory made possible by a recent rekindling of interest in the R. C. James inequalities for super-reflexive spaces. When the James inequalities are combined with Young-Stieltjes integration for the spaces $V_{p}(𝕋)$ of functions having bounded p-variation, it transpires that every trigonometrically well-bounded operator on a super-reflexive space X has a norm-continuous $V_{p}(𝕋)$-functional calculus for a range of values of p > 1, and we investigate the ways this outcome logically simplifies and simultaneously expands the structure theory, Fourier analysis, and operator ergodic theory of trigonometrically well-bounded operators on X. In particular, on a super-reflexive space X (but not on a general relexive space) a theorem of Tauberian type holds: the (C,1) averages in (*) corresponding to a trigonometrically well-bounded operator U can be replaced by the set of all the rotated ergodic Hilbert averages of U, which, in fact, is a precompact set relative to the strong operator topology. This circle of ideas is facilitated by the development of a convergence theorem for nets of spectral integrals of $V_{p}(𝕋)$-functions. In the Hilbert space setting we apply the foregoing to the operator-weighted shifts which are known to provide a universal model for trigonometrically well-bounded operators on Hilbert space.
3
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An $M_{q}(𝕋)$-functional calculus for power-bounded operators on certain UMD spaces

64%
Berkson E., Gillespie T.
Studia Mathematica
|
2005
|
tom 167
|
nr 3
245-257
EN
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.
4
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Spectral decompositions, ergodic averages, and the Hilbert transform

64%
Berkson E., Gillespie T.
Studia Mathematica
|
2001
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tom 144
|
nr 1
39-61
EN
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.
5
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Spectral decompositions and harmonic analysis on UMD spaces

64%
Berkson E., Gillespie T. A.
Studia Mathematica
|
1994-1995
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tom 112
|
nr 1
13-49
EN
We develop a spectral-theoretic harmonic analysis for an arbitrary UMD space X. Our approach utilizes the spectral decomposability of X and the multiplier theory for $L_X^p$ to provide on the space X itself analogues of the classical themes embodied in the Littlewood-Paley Theorem, the Strong Marcinkiewicz Multiplier Theorem, and the M. Riesz Property. In particular, it is shown by spectral integration that classical Marcinkiewicz multipliers have associated transforms acting on X.
6
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$L^{p}$-Multiplier transference induced by representations in Hilbert space

51%
Berkson E., Gillespie T. A., Muhly P. S.
Studia Mathematica
|
1989
|
tom 94
|
nr 1
51-61
7
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The spectral decomposition of weighted shifts and the $A^p$ condition

51%
Berkson E., Gillespie T. A.
Colloquium Mathematicum
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1990
|
tom 60-61
|
nr 2
507-518
8
Content available remote

The hermitian operators on some Banach spaces

38%
Berkson E., Sourour A.
Studia Mathematica
|
1974-1975
|
tom 52
|
nr 1
33-41
9
Content available remote

Restrictions from $ℝ^{n}$ to $ℤ^{n}$ of weak type (1,1) multipliers

32%
Asmar N., Berkson E., Bourgain J.
Studia Mathematica
|
1994
|
tom 108
|
nr 3
291-299
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