Hervé Queffélec

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

Approximation numbers of composition operators on the H2 space of Dirichlet series

By a theorem of Gordon and Hedenmalm, φ generates a bounded composition operator on the Hilbert space H2 of Dirichlet series ∑nbnn−s with square-summable coefficients bn if and only if φ(s)=c0s+ψ(s), where c0 is a nonnegative integer and ψ a Dirichlet series with the following mapping properties: ψ maps the right half-plane into the half-plane Res>1/2 if c0=0 and is either identically zero or maps the right half-plane into itself if c0 is positive. It is shown that the nth approximation numbers of bounded composition operators on H2 are bounded below by a constant times rn for some 0 0 when c0 is positive. Both results are best possible. The case when c0=0, ψ is bounded and smooth up to the boundary of the right half-plane, and supu2061Reψ=1/2, is discussed in depth; it includes examples of non-compact operators as well as operators belonging to all Schatten classes Sp. For φ(s)=c1+∑j=1dcqjqj−s with qj independent integers, it is shown that the nth approximation number behaves as n−(d−1)/2, possibly up to a factor (logu2061n)(d−1)/2. Estimates rely mainly on a general Hilbert space method involving finite linear combinations of reproducing kernels. A key role is played by a recently developed interpolation method for H2 using estimates of solutions of the ∂¯ equation. Finally, by a transference principle from H2 of the unit disc, explicit examples of compact composition operators with approximation numbers decaying at essentially any sub-exponential rate can be displayed.

Some revisited results about composition operators on Hardy spaces

We generalize, on one hand, some results known for composition operators on Hardy spaces to the case of Hardy-Orlicz spaces H � : construction of a slow Blaschke product giving a non-compact composition operator on H � ; construction of a surjective symbol whose composition operator is compact on Hand, moreover, is in all the Schatten classes Sp(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. Mathematics Subject Classification. Primary: 47B33 - Secondary: 47B10

Full length article: On approximation numbers of composition operators

We show that the approximation numbers of a compact composition operator on the Hardy space H^2 or on the weighted Bergman spaces B@a of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they cannot decay more rapidly than exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bound and explicit an example.

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.

Compact composition operators on Bergman-Orlicz spaces

We construct an analytic self-map of the unit disk and an Orlicz functionfor which the composition operator of symbol is compact on the Hardy-Orlicz space H � , but not on the Bergman-Orlicz space B � . For that, we first prove a Carleson embedding theorem, and then characterize the compact- ness of composition operators on Bergman-Orlicz spaces, in terms of Carleson function (of order 2). We show that this Carleson function is equivalent to the Nevanlinna counting function of order 2. Mathematics Subject Classification. Primary: 47B33 - Secondary: 30D50; 30D55; 46E15

Decay rates for approximation numbers of composition operators

A general method for estimating the approximation numbers of composition operators on the Hardy space H2, using finite-dimensional model subspaces, is studied and applied in the case when the symbol of the operator maps the unit disc to a domain whose boundary meets the unit circle at just one point. The exact rate of decay of the approximation numbers is identified when this map is sufficiently smooth at the point of tangency; it follows that a composition operator with any prescribed slow decay of its approximation numbers can be explicitly constructed. Similarly, an asymptotic expression for the approximation numbers is found when the mapping has a sharp cusp at the distinguished boundary point. Precise asymptotic estimates in the intermediate cases, including that of maps with a corner at the distinguished boundary point, are also established.

Approximation numbers of composition operators on H p

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

On some random thin sets of integers

We show how different random thin sets of integers may have behaviour. First using a recent deviation inequality of Boucheron. different behaviour. First, using a recent deviation inequality of Boucheron, Lugosi and Massart, we give a simpler proof of one of our results in Some new thin sets of integers in harmonic analysis, Journal d Analyse Mathematique 86 (2002), 105-138, namely that there exist 4/3-Rider sets which are sets of uniform convergence and Λ(q)-sets for all q 4/3, the p-Rider sets which we had constructed in that paper are almost surely not of uniform convergence.

Estimate for Norm of a Composition Operator on the Hardy–Dirichlet Space

By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on