Approximation numbers of weighted composition operators

We study the approximation numbers of weighted composition operators $f\mapsto w\cdot(f\circ\varphi)$ on the Hardy space $H^2$ on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers are derived. For the special class of weighted lens map composition operators with specific weights, we show how much the weight $w$ can improve the decay rate of the approximation numbers, and give sharp upper and lower bounds. These examples are motivated from applications to the analysis of relative commutants of special inclusions of von Neumann algebras appearing in quantum field theory (Borchers triples).Comment: 35 pages, no figures. Some typos removed, minor improvements in presentation, updated reference

Spectra of some invertible weighted composition operators on Hardy and weighted Bergman spaces in the unit ball

In this paper, we investigate the spectra of invertible weighted composition operators with automorphism symbols, on Hardy space $H^2(\mathbb{B}_N)$ and weighted Bergman spaces $A_\alpha^2(\mathbb{B}_N)$, where $\mathbb{B}_N$ is the unit ball of the $N$-dimensional complex space. By taking $N=1$, $\mathbb{B}_N=\mathbb{D}$ the unit disc, we also complete the discussion about the spectrum of a weighted composition operator when it is invertible on $H^2(\mathbb{D})$ or $A_\alpha^2(\mathbb{D})$.Comment: 23 Page

Mini-Workshop: Operators on Spaces of Analytic Functions

The major topics discussed in this workshop were invariant subspaces of linear operators on Banach spaces of analytic functions, the ideal structure of H ∞ , asymptotics for condition numbers of large matrices, and questions related to composition operators, frequently hypercyclic operators, subnormal operators and generalized Cesàro operators. A list of open problems raised at this workshop is also included

Norms of some operators between weighted-type spaces and weighted Lebesgue spaces

We calculate the norms of several concrete operators, mostly of some integral-type ones between weighted-type spaces of continuous functions on several domains. We also calculate the norm of an integral-type operator on some subspaces of the weighted Lebesgue spaces

Hilbert Modules and Complex Geometry

The major topics discussed in the workshop were Hilbert modules of analytic functions on domains in Cn, Toeplitz and Hankel operators, reproducing kernel Hilbert spaces and multiplier algebras, the interplay of complex geometry and operator theory, non-commutative function theory and operator theory, Hilbert bundles on symmetric spaces

Intertwining relations, commutativity and orbits

The density of orbits and commutativity up to a factor of bounded linear operators have become of great interest for Operator Theorists during the last decades. This interest comes from the relationship that exists between the study of orbits and spectral properties of linear operators on Banach spaces and the Invariant Subspace Problem. As a consequence, the research on the Theory of Hyperclicity has increased considerably. In this manuscript, we characterize the hypercyclicity of the Cesàro means of higher-order on Banach spaces. We prove some sufficient conditions on the extended-spectrum of a bounded linear operator that guarantee its non convex-cyclicity. The notion of convex-cyclicity, was introduced by H. Rezaei in 2013 (see \cite{reza}). It is a sufficient condition for cyclicity and a necessary condition for hypercyclicity. We characterize the hypercyclicity of operators commuting up to a factor with the differentiation operator in the space of entire functions equipped with the topology of uniform convergence for compact sets. Our results are an extension of some of the most classical results related to the differentiation operator, that is, the ones of G. Godefroy and J. H. Shapiro \cite{Godefroy1991}, and R. Aron and D. Markose \cite{AronMarkose2004}. Next, we consider some particular operators, such as composition operators in weighted Hardy spaces. These operators have been studied intensely by several mathematicians in the Hardy space, see the recent of \cite{leon4}. Although we know fewer things about these operators in weighted Hardy spaces, we calculated the extended-spectrum of composition operators that are induced by a bilinear transformation that fixes an interior point of the unit disk and an exterior one of its closure. Namely, we treat the elliptic, loxodromic cases and a hyperbolic subcase. Finally, we continue to the study of the more general unbounded operators. After the paper of von Neumann \cite{von-Neumman-historic-Fuglede}, commutativity and intertwining relations of unbounded operators have been developed by many mathematicians. Among these mathematicians, we state Bent Fuglede whose Theorem was an improvement of the Spectral Theorem for Normal Operators. We show a new version of the Fuglede Theorem for unbounded normal operators

Operators on some analytic function spaces and their dyadic counterparts

In this thesis we consider several questions on harmonic and analytic functions spaces and some of their operators. These questions deal with Carleson-type measures in the unit ball, bi-parameter paraproducts and multipliers problem on the bitorus, boundedness of the Bergman projection and analytic Besov spaces in tube domains over symmetric cones. In part I of this thesis, we show how to generate Carleson measures from a class of weighted Carleson measures in the unit ball. The results are used to obtain boundedness criteria of the multiplication operators and Ces`aro integral-type operators between weighted spaces of functions of bounded mean oscillation in the unit ball. In part II of this thesis, we introduce a notion of functions of logarithmic oscillation on the bitorus. We prove using Cotlar’s lemma that the dyadic version of the set of such functions is the exact range of symbols of bounded bi-parameter paraproducts on the space of functions of dyadic bounded mean oscillation. We also introduce the little space of functions of logarithmic mean oscillation in the same spirit as the little space of functions of bounded mean oscillation of Cotlar and Sadosky. We obtain that the intersection of these two spaces of functions of logarithmic mean oscillation and L1 is the set of multipliers of the space of functions of bounded mean oscillation in the bitorus. In part III of this thesis, in the setting of the tube domains over irreducible symmetric cones, we prove that the Bergman projection P is bounded on the Lebesgue space Lp if and only if the natural mapping of the Bergman space Ap0 to the dual space (Ap) of the Bergman space Ap, where 1 p + 1 p0 = 1, is onto. On the other hand, we prove that for p > 2, the boundedness of the Bergman projection is also equivalent to the validity of an Hardy-type inequality. We then develop a theory of analytic Besov spaces in this setting defined by using the corresponding Hardy’s inequality. We prove that these Besov spaces are the exact range of symbols of Schatten classes of Hankel operators on the Bergman space A2