arXiv subjects
Publications and source records attributed to Daniel Li.
We give general estimates for the approximation numbers of composition operators on the Hardy space on the ball $B\_d$ and the polydisk $D^d$ .
We give estimates for the approximation numbers of composition operators on the Hp spaces, 1 $\le$ p \textless{} $\infty$.
We show that the decay of approximation numbers of compact composition operators on the Dirichlet space $\mathcal{D}$ can be as slow as we wish, which was left open in the cited work. We also prove the optimality of a result of O.~El-Fallah, K.~Kellay, M.~Shabankhah and A.~Youssfi on boundedness on $\mathcal{D}$ of self-maps of the disk all of whose powers are norm-bounded in $\mathcal{D}$.
For approximation numbers $a_n (C_ϕ)$ of composition operators $C_ϕ$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol $ϕ$ of uniform norm $< 1$, we prove that $\lim_{n \to \infty} [a_n (C_ϕ)]^{1/n} = \e^{- 1/ \capa [ϕ(\D)]}$, where $\capa [ϕ(\D)]$ is the Green capacity of $ϕ(\D)$ in $\D$. This formula holds also for $H^p$ with $1 \leq p < \infty$.
We study the decay of approximation numbers of compact composition operators on the Dirichlet space. We give upper and lower bounds for these numbers. In particular, we improve on a result of O. El-Fallah, K. Kellay, M. Shabankhah and A. Youssfi, on the set of contact points with the unit circle of a compact symbolic composition operator acting on the Dirichlet space D. We extend their results in two directions: first, the contact only takes place at the point 1. Moreover, the approximation numbers of the operator can be arbitrarily sub-exponentially small.
In this paper, we prove that for every compact set of the unit disk of logarithmic capacity 0, there exists a Schur function both in the disk algebra and in the Dirichlet space such that the associated composition operator is in all Schatten classes (of the Dirichlet space), and for which the set of points whose image touches the unit circle is equal to this compact set. We show that for every bounded composition operator on the Dirichlet space and for every point of the unit circle, the logarithmic capacity of the set of point having this point as image is 0. We show that every compact composition operator on the Dirichlet space is compact on the gaussian Hardy-Orlicz space; in particular, it is in every Schatten class on the usual Hilbertian Hardy space. On the other hand, there exists a Schur function such that the associated composition operator is compact on the gaussian Hardy-Orlicz space, but which is not even bounded on the Dirichlet space. We prove that the Schatten classes on the Dirichlet space can be separated by composition operators. Also, there exists a Schur function such that the associated composition operator is compact on the Dirichlet space, but in no Schatten class.
We prove that, for every $α> -1$, the pull-back measure $ϕ({\cal A}_α)$ of the measure $d{\cal A}_α(z) = (α+ 1) (1 - |z|^2)^α\, d{\cal A} (z)$, where ${\cal A}$ is the normalized area measure on the unit disk $\D$, by every analytic self-map $ϕ\colon \D \to \D$ is not only an $(α+ 2)$-Carleson measure, but that the measure of the Carleson windows of size $\eps h$ is controlled by $\eps^{α+ 2}$ times the measure of the corresponding window of size $h$. This means that the property of being an $(α+ 2)$-Carleson measure is true at all infinitesimal scales. We give an application by characterizing the compactness of composition operators on weighted Bergman-Orlicz spaces.
We give estimates for the approximation numbers of composition operators on $H^2$, in terms of some modulus of continuity. For symbols whose image is contained in a polygon, we get that these approximation numbers are dominated by $\e^{- c \sqrt n}$. When the symbol is continuous on the closed unit disk and has a domain touching the boundary non-tangentially at a finite number of points, with a good behavior at the boundary around those points, we can improve this upper estimate. A lower estimate is given when this symbol has a good radial behavior at some point. As an application we get that, for the cusp map, the approximation numbers are equivalent, up to constants, to $\e^{- c \, n / \log n}$, very near to the minimal value $\e^{- c \, n}$. We also see the limitations of our methods. To finish, we improve a result of O. El-Fallah, K. Kellay, M. Shabankhah and H. Youssfi, in showing that for every compact set $K$ of the unit circle $\T$ with Lebesgue measure 0, there exists a compact composition operator $C_ϕ\colon H^2 \to H^2$, which is in all Schatten classes, and such that $ϕ= 1$ on $K$ and $|ϕ| < 1$ outside $K$.
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 $H^ψ$ and $B^ψ$, 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.
We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces $\mathfrak{B}_α$ of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they grow at least exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bounds and explicit an example.
It is known, from results of B. MacCluer and J. Shapiro (1986), that every composition operator which is compact on the Hardy space $H^p$, $1 \leq p < \infty$, is also compact on the Bergman space ${\mathfrak B}^p = L^p_a (\D)$. In this survey, after having described the above known results, we consider Hardy-Orlicz $H^Ψ$ and Bergman-Orlicz ${\mathfrak B}^Ψ$ spaces, characterize the compactness of their composition operators, and show that there exist Orlicz functions for which there are composition operators which are compact on $H^Ψ$ but not on ${\mathfrak B}^Ψ$.
We construct an analytic self-map $ϕ$ of the unit disk and an Orlicz function $Ψ$ for which the composition operator of symbol $ϕ$ is compact on the Hardy-Orlicz space $H^Ψ$, but not compact on the Bergman-Orlicz space ${\mathfrak B}^Ψ$. For that, we first prove a Carleson embedding theorem, and then characterize the compactness 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.
We study the canonical injection from the Hardy-Orlicz space $H^Ψ$ into the Bergman-Orlicz space ${\mathfrak B}^Ψ$.
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 randomly construct various subsets $Λ$ of the integers which have both smallness and largeness properties. They are small since they are very close, in various meanings, to Sidon sets: the continuous functions with spectrum in $Λ$ have uniformly convergent series, and their Fourier coefficients are in $\ell_p$ for all $p>1$; moreover, all the Lebesgue spaces $L^q_Λ$ are equal for $q<+\infty$. 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 non separable. So these sets are very different from the thin sets of integers previously known.
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 $(V)$.