-Sidon sets

The aim of this paper is to prove that for every , every -Rider set is a -Sidon set for all {\frac p{2-p}}\cdot$"> This gives some positive answers for the union problem of -Sidon sets. We also obtain some results on the behavior of the Fourier coefficient of a measure with spectrum in a -Rider set.

     

Operator ideal norms on

Let be a real number such that and its conjugate exponent . We prove that for an operator defined on with values in a Banach space, the image of the unit ball determines whether belongs to any operator ideal and its operator ideal norm. We also show that this result fails to be true in the remaining cases of . Finally we prove that when the result holds in finite dimension, the map which associates to the image of the unit ball the operator ideal norm is continuous with respect to the Hausdorff metric.

     

Nevanlinna counting function and Carleson function of analytic maps

We show that the maximal Nevanlinna counting function and the Carleson function of analytic self-maps of the unit disk are equivalent, up to constants.     

On the structure of spaces of uniformly convergent Fourier series

We first give some new examples of translation invariant subspaces of C or U without local unconditional structure. In the second part, we prove that U and U ⁺ do not have the Gordon–Lewis property. In the third part, we show that absolutely summing operators from U to a K-convex space are compact. As a consequence, U and U ⁺ are not isomorphic. At last, we prove that U and U ⁺ do not have the Daugavet property.     

Infinitesimal Carleson Property for Weighted Measures Induced by Analytic Self-Maps of the Unit Disk

We prove that, for every $\alpha > -1$ , the pull-back measure $\varphi ({\mathcal A }_\alpha )$ of the measure $d{\mathcal A }_\alpha (z) = (\alpha + 1) (1 - |z|^2)^\alpha \, d{\mathcal A } (z)$ , where ${\mathcal A }$ is the normalized area measure on the unit disk $\mathbb D $ , by every analytic self-map $\varphi :\mathbb D \rightarrow \mathbb D $ is not only an $(\alpha \,{+}\, 2)$ -Carleson measure, but that the measure of the Carleson windows of size $\varepsilon h$ is controlled by $\varepsilon ^{\alpha + 2}$ times the measure of the corresponding window of size $h$ . This means that the property of being an $(\alpha + 2)$ -Carleson measure is true at all infinitesimal scales. We give an application by characterizing the compactness of composition operators on weighted Bergman–Orlicz spaces.     

Approximation numbers of weighted composition operators

We study the approximation numbers of weighted composition operators $f?w?(f°\phi )$ on the Hardy space $H^2$ on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers are derived. For the special class of weighted lens map composition operators with specific weights, we show how much the weight w can improve the decay rate of the approximation numbers, and give sharp upper and lower bounds. These examples are motivated from applications to the analysis of relative commutants of special inclusions of von Neumann algebras appearing in quantum field theory (Borchers triples).     

Some new thin sets of integers in harmonic analysis

We randomly construct various subsets A of the integers which have both smallness and largeness properties. They are small since they are very close, in various senses, to Sidon sets: the continuous functions with spectrum in Λ have uniformly convergent series, and their Fourier coefficients are in ℓ_p for all p > 1; moreover, all the Lebesgue spaces L _Λ ^q are equal forq < +∞. On the other hand, they are large in the sense that they are dense in the Bohr group and that the space of the bounded functions with spectrum in Λ is nonseparable. So these sets are very different from the thin sets of integers previously known.     

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that _limn→∞?[_an(_Cφ)]^1/n=e^{−1/Cap[φ(D)]}${\lim }_{n\to \infty }?{[a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}[\phi (\mathbb{D})]}$, where Cap[φ(D)]$\mathrm{Cap}[\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.     

Some new properties of composition operators associated with lens maps

We give examples of results on composition operators connected with lens maps. The first two concern the approximation numbers of those operators acting on the usual Hardy space H ². The last ones are connected with Hardy-Orlicz and Bergman-Orlicz spaces ${H^\psi }$ and ${B^\psi }$ , and provide a negative answer to the question of knowing if all composition operators which are weakly compact on a non-reflexive space are norm-compact.