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Program and Book of Abstracts Abstract List Alexandru ALEMAN University of Lund Spectrum of the Hilbert matrix on Banach spaces of analytic functions. The talk presents a unified approach to the spectrum of the operator induced by the Hilbert matrix acting on Taylor coefficients based on conformal invariance and the properties of the so-called reduced Hilbert matrix. It leads to a complete description of the spectrum for a large class of Banach spaces on the unit disc containing Hardy, Bergman and Dirichlet spaces. The method extends also for some spaces which lack conformal invariance, for example, `p; p > 1. This is a report on work in progress, joint with A. Siskakis and D. Vukotic. Anton BARANOV St. Petersburg State University Spectral theory of rank one perturbations of normal operators. We use a functional model for rank one perturbations of compact normal operators to study their spectral properties. In particular, we discuss completeness of a rank one perturbation and of its adjoint as well as the possibility of the spectral synthesis, i.e., reconstruction of any invariant subspace from the eigenvectors it contains. The functional model acts in some relatively new class of spaces of entire functions which generalize the de Branges spaces. Fr´ed´ericBAYART Universit´eClermont-Auvergne On the interpolation of Hardy spaces of Dirichlet series. 1 2 In this talk, we will prove that, for the usual methods of interpolation, the interpolated space between two Hardy spaces of Dirichlet series is not the Hardy space of Dirichlet series we would expect (based on a joint work with M. Mastylo). Yurii BELOV St. Petersburg State University Density of complete and minimal systems of time frequency shifts of Gaussians. We study the upper and the lower densities of complete and minimal Gabor Gaussians systems. In contrast to the classical lattice case when they are both equal to 1, we prove 1 that the lower density may reach 0 while the upper density may vary at least from π to e. (Joint work with A. Borichev and A. Kuznetsov). This work is supported by the Russian Science Foundation grant 19-11-00058. Alexander BORICHEV University Aix-Marseille Taylor series with random and pseudo-random coefficients and the Wiener spec- trum. Given a sequence (an) we discus different conditions which guarantee the angular equidistri- P n bution of the zeros of the Taylor series anz =n!. Filippo BRACCI Universit`adi Roma 'Tor Vergata' The slope of orbits of semigroups of holomorphic self-maps of the unit disc. Let D be a simply connected domain in the complex plane, R a continuous curve in D landing at the boundary of D and f a Riemann map from the unit disc to D. It is well known that f −1(R) lands to a point x on the boundary of the unit disc. The question this talks concerns on is: does one can find (useful) geometric conditions on D that guarantees that f −1(R) converges non-tangentially (or tangentially) to x? I will provide a necessary 3 and sufficient answer to this question based on recent works with H. Guassier, M. Contreras, S. Diaz-Madrigal and A. Zimmer. Such a question is particularly interesting in studying dynamics of orbits of continuous semigroups of holomorphic self-maps of the unit disc (or more generally iterates of holomorphic selfmaps of the unit disc). Indeed, in such case, if there is no fixed points in the unit disc, the orbits are all landing at a point of the boundary (the Denjoy-Wolff point) and the question is about the type of convergence. Every continuous semigroup of holomorphic self-maps of the unit disc admits an essentially unique "holomorphic model", that it, it is conjugated via a Riemann map to the dynamical system z 7! z + it; t ≥ 0, where z lives in a simply connected domain D starlike at infinity (image of the intertwining Riemann map). I will explain that the orbit converge non-tangentially to the Denjoy-Wolff point if and only if D is "quasi-symmetric" with respect to one { and hence any { vertical axis. This is also equivalent to the condition that one { and hence every { orbit of the semigroup is a "quasi-geodesic" in the sense of Gromov. A similar result holds for "tangential convergence". The proof is based on new interplays between the Euclidean geometry of the domain D and the hyperbolic geometry, via the Gromov hyperbolicity theory and new results of localization of the hyperbolic distance. This also brings interesting simple consequences: for instance, if D contains a vertical angle, then the convergence is non-tangential, and this allows to construct "pathological" examples of semigroups with orbits that converge non non-tangentially but not tangentially, or non- tangentially but oscillating to the Denjoy-Wolff point. Isabelle CHALENDAR Universit´eParis-Est | Marne-la-Vall´ee In Koenigs' footsteps: diagonalization of composition operators. Let D be the open unit disc in the complex plane. Let ' ! D be a holomorphic map with a fixed point α 2 D such that 0 ≤ '0(α) < 1. We show that the spectrum of the composition 0 n operator C' on the Fr´echet space Hol(D) is f0g [ f' (α) : n = 0; 1;::: g and its essential spectrum is reduced to f0g. This contrasts the situation where a restriction of C' to Banach spaces such as H2(D) is considered. The proofs are based on explicit formulae of the spectral projections associated with the point spectrum found by Koenig. Finally, as a byproduct, we obtain information on the spectrum for bounded composition operators induced by a Schr¨odersymbol on arbitrary Banach spaces of holomorphic functions. 4 Manuel CONTRERAS University of Seville On the rate of convergence of semigroups of holomorphic functions. Continuous one-parameter semigroups of holomorphic self-maps of the unit disc D in the complex plane C have been a subject of study since early 1900's, both for their intrinsic interest in complex analysis and for applications. They were first considered in the 1930's by J. Wolff, although it was only with a paper of E. Berkson and H. Porta in the 1970's that the modern study of semigroups in D initiated. Since their work, interest in semigroups in D has expanded due to various applications in physics and biology and their connections to composition operators and Loewner's theory. Given a semigroup of holomorphic functions in the unit disc, ('t), different from an elliptic group, the Denjoy-Wolff theorem shows that there exists a point τ 2 D such that ('t) converges uniformly on compact subsets of D to the constant map D 3 z 7! τ as t tends to +1. Fixed z 2 D, in this talk we will analyze the rate of convergence of 't(z) to τ, as t tends to +1. We will use techniques coming from harmonic measures. These results will appear in a joint paper with D. Betsakos and S. D´ıaz-Madrigal. Eva GALLARDO-GUTIERREZ´ Complutense University, Madrid On Bishop operators: invariant subspaces and spectral decompositions. Bishop's operator arose in the fifties as possible candidates for being counterexamples to the Invariant Subspace Problem. Several authors have addressed the problem of finding invariant subspaces for some of these operators; but still the general problem is open. In this talk, we shall discuss about recent results on the existence of invariant subspaces an their relation with spectral decompositions for the Bishop operators. (Joint work with F. Chamizo, M. Monsalve-L´opez and A. Ubis) Karlheinz GROCHENIG¨ University of Wien Sampling, Marcinkiewicz-Zygmund sets, approximation, and quadrature rules. Suppose you are given n samples of a function f on the torus. What can you say about f? R 1 How well can you approximate f or the integral 0 f(x) dx? The key notion to answer this question is the notion of Marcinkiewicz-Zygmund (MZ) sets [1; 2]. A MZ-set X = (Xn)n2N 5 for the torus is a sequence of finite sets Xn such that 2 X 2 2 Akpk2 ≤ jp(x)j ≤ Bkpk2; p 2 Tn; x2Xn where Tn denotes the subspace of trigonometric polynomials of degree n and A; B > 0 are two constants independent of n. To produce a good approximation of a smooth function f from its samples fjXn on an MZ-set, one first constructs a trigonometric polynomial pn 2 Tn that approximates the given samples optimally. Numerically, this amounts to solving a least squares problem. For f in the Sobolev space Hσ, one can then prove a convergence rate −σ+1=2 kf − pnk = O(n ): This also implies an error estimate for quadrature rule that is associated to every MZ-set. Using the techniques from [3] one can prove similar results for functions on a bounded domain in Rd, where the trigonometric system is replaced by the eigenfunctions of the Laplacian, or even more generally on compact Riemannian manifolds. Bibliography [1] A. Cohen and G. Migliorati. Optimal weighted least-squares methods. SIAM J. Comput. Math., 3:181-203, 2017. [2] F. Filbir and H. N. Mhaskar. Marcinkiewicz-Zygmund measures on manifolds. J. Complexity, 27(6):568-96, 2011. [3] K. Gr¨ochenig. Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type. Math. Comp., 68(226):749-765, 1999. Pavel MOZOLYAKO University of Bologna and Michigan State University Two-weighted Hardy inequality on the tree and bitree. We consider a two-weighted trace inequality for the Hardy operator on the dyadic tree and on the bitree (a graph that can be interpreted as a collection of dyadic subrectangles of [0; 1]2). We show how this inequality is connected to different Carleson-type embeddings for Hardy, Dirichlet and Bergman spaces on the disc (and bidisc as well).
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