Daniel Li

Composition operators on Hardy-Orlicz spaces

We investigate composition operators on Hardy-Orlicz spaces when the Orlicz function Ψ grows rapidly: compactness, weak compactness, to be psumming, order bounded, . . . , and show how these notions behave according to the growth of Ψ. We introduce an adapted version of Carleson measure. We construct various examples showing that our results are essentially sharp. In the last part, we study the case of Bergman-Orlicz spaces. Mathematics Subject Classification. Primary: 47 B 33 – 46 E 30; Secondary: Key-words. Bergman-Orlicz space – Carleson measure – composition operator – Hardy-Orlicz space

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 ℓp for all p > 1; moreover, all the Lebesgue spaces LΛq 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 criterion of weak compactness for operators on subspaces of Orlicz spaces

We give a criterion of weak compactness for the operators on the Morse-Transue space M Ψ , the subspace of the Orlicz space L Ψ generated by L ∞ .

Approximation numbers of composition operators on H p

Abstract give estimates for the approximation numbers of composition operators on the Hp spaces, 1 ≤ p < ∞