arXiv subjects
Publications and source records attributed to Pascal Lefèvre.
We characterize the compactness of composition operators; in term of generalized Nevanlinna counting functions, on a large class of Hilbert spaces of analytic functions, which can be viewed between the Bergman and the Dirichlet spaces
We generalize, on one hand, some results known for composition operators on Hardy spaces to the case of Hardy-Orlicz spaces $H^Ψ$: construction of a "slow" Blaschke product giving a non-compact composition operator on $H^Ψ$; construction of a surjective symbol whose composition operator is compact on $H^Ψ$ and, moreover, is in all the Schatten classes $S_p (H^2)$, $p > 0$. On the other hand, we revisit the classical case of composition operators on $H^2$, giving first a new, and simplier, characterization of closed range composition operators, and then showing directly the equivalence of the two characterizations of membership in the Schatten classes of Luecking and Luecking and Zhu.
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.
We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space $c_0$.
We give new proofs that some Banach spaces have Pełczyński's property $(V)$.
We investigate the behavior of some thin sets of integers defined through random trigonometric polynomial when one replaces Gaussian or Rademacher variables by p-stable ones, with 1 < p < 2. We show that in one case this behavior is essentially the same as in the Gaussian case, whereas in another case, this behavior is entirely different.
We investigate composition operators on Hardy-Orlicz spaces when the Orlicz function $Ψ$ grows rapidly: compactness, weak compactness, to be $p$-summing, 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.