Talks

In 1966 D. Newman and H. Shapiro posed the following problem. Let $G$ be a function from Fock space $\mathcal{F}$ and such that $e^{zw}G\in\mathcal{F}$ for any $w\in\mathbb{C}$. Is it true that $$Span{FG; FG ∈ F} = Span{e^{wz}G : w ∈ C}?$$ Recently the author (joint with A.Borichev) has constructed a counterexample to this conjecture. On the other hand, we are able to show that if $G$ satisfies some regularity conditions, then conjecture holds. These results are clоsеly connected to some spectral synthesis problems in Fock space.

In this talk we present a survey of results on the structure of contractive and bi-contractive projections on spaces of continuous functions. We also give a characterization for the bi-contractive projections on spaces of continuous functions defined on a compact Hausdorff space and with values in a separable Hilbert space.

Contractive projections and Bi-contractive Projections on Spaces of Vector Valued Continuous Functions

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The concept of a Nevanlinna domain is the special analytic characteristic of bounded simply connected domains in the complex plane. It plays a crucial role in recent advances in problems of uniform approximation of functions on compact sets in the plane by polynomial solutions of elliptic equations with constant complex coefficients. The properties of Nevanlinna domains deeply relate with the properties of bounded univalent functions belonging to model spaces (that is to subspaces of the Hardy space $H^2$, which are invariant with respect to the backward shift operator). One of the most interesting problem concerning Nevanlinna domains is the following question which was posed in the early 2000s: how large (in the sense of dimension theory) there can be boundaries of Nevanlinna domains? In the talk it is planned to explain the history of this question and present it’s final solution which was recently obtained by Yurii Belov, Alexander Borichev and the author.

In this talk we consider systems described by the heat equation on the interval $[0,\pi]$ with $L^2$ boundary controls and study the reachable space at some instant $\tau>0$. The main results assert that this space is generally sandwiched between two Hilbert spaces of holomorphic functions defined on a square in the complex plane and which has $[0,\pi]$ as one of the diagonals. More precisely, in the case of Dirichlet boundary controls acting at both ends we prove that the reachable space contains the Hardy-Smirnov space and it is contained in the Bergman space associated to the above mentioned square. For this we discuss the (explicit) operator associating with both boundary control functions (the so-called input) the corresponding state of the system. This operator is commonly called the input-to-state map. A key step, based on the Poisson summation formula, is the representation of this operator as a convolution with a sum of Gaussians. As it turns out, only two terms of this sum corresponding to both boundary controls need to be considered. These leading terms give rise to integral transforms associated with the heat kernel (or weighted Laplace transforms on sectors) which have been investigated by Aikawa, Hayashi and Saitoh. Another key ingredient is given by Riesz bases for Hardy-Smirnov spaces in polygons which have been discussed by Levin and Lyubarskii.

The Hardy spaces of Dirichlet series $\mathcal{H}^p$ ($p\geq 1$) have been studied by Hedenmalm, Lindqvist and Seip when $p=2$ and by Bayart for the general values of $p$. We introduce the Orlicz version of Hardy spaces of Dirichlet series $\mathcal{H}^\psi$. We focus on the case $\psi=\psi_q(t)=\exp(t^q)-1$ and we compute the abscissa of convergence for these spaces, obtaining values interpolating the ones for $\mathcal{H}^p$ and $\mathcal{H}^\infty$. This solves a problem raised by Hedenmalm in 2002. This talk is a complement to the lectures of F. Bayart. Joint work with M. Bailleul and L. Rodr\'\i guez-Piazza.

We consider analytic composition operators in the unit disc. We show that they exhibit quite strong rigidity of non-compact behaviour on Hardy and other function spaces. In particular, we characterize strictly singular and l^2-singular composition operators on H^p. We also consider generalizations of these results.

We introduce and study spaces of entire functions in several complex variables whose restriction to a strongly pseudoconvex manifold satisfy some integrability conditions. For these spaces we prove Paley-Wiener type theorems and some other structural properties. These spaces are a natural generalisation of the classical Paley—Wiener space in variable but also define a new class of spaces in the complex plane. We determine some of their structural properties and study the characterisation of sampling, interpolating and complete interpolating sequences. This is report on joint work with A. Monguzzi and S. Salvatori.

We discuss recent work on the two weight Tb theorem. This is joint work with Chun-Yen Shen and Ignacio Uriarte-Tuero.

We present recent results with Bénéteau and Manolaki with respect to the invariant subspaces generated by a function f in a function space. The information on such spaces is contained on a family of polynomials that we call optimal approximants. In this work, we provide a method for finding a closed formula for all such approximants for a fixed function f, with some assumptions.