Zero sequences, factorization and sampling measures for weighted Bergman spaces

We show that the Korenblum maximum (domination) principle is valid for weighted Bergman spaces Awp with arbitrary (non-negative and integrable) radial weights w in the case 1≤p<∞. We also notice that in every weighted Bergman space the supremum of all radii for which the principle holds is strictly smaller than one. Under the mild additional assumption lim infr→0+w(r)>00$$\\end{document}]]>, we show that the principle fails whenever 0<p<1.

The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces $$A_{\alpha }^p$$ with $$p \in (0, \infty )$$ and $$ \alpha \in (-1, \infty )$$ . In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

On the Bergman space in the unit disk, we study the intertwining relation for Volterra type operators, whose intertwining operator is a composition operator. We also investigate the “compact” intertwining relations for Volterra type operators. As obvious consequences, the essential commutativity of Volterra type and composition operators are characterized. At the end of the paper, we find a new connection between the Bergman space and little Bloch space through this essential commutativity.

Let \Omega be a subdomain of \mathbb{C} and let \mu be a positive Borel measure on \Omega . In this paper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operators T_\mu acting on Bergman spaces on \Omega . Let (\lambda _n(T_\mu)) be the decreasing sequence of the eigenvalues of T_\mu , and let \rho be an increasing function such that \rho (n)/n^A is decreasing for some A&gt;0 . We give an explicit necessary and sufficient geometric condition on \mu in order to have \lambda _n(T_\mu)\asymp 1/\rho (n) . As applications, we consider composition operators C_\varphi , acting on some standard analytic spaces on the unit disc \mathbb{D} . First, we give a general criterion ensuring that the singular values of C_\varphi satisfy s_n(C_\varphi ) \asymp 1/\rho(n) . Next, we focus our attention on composition operators with univalent symbols, where we express our general criterion in terms of the harmonic measure of \varphi (\mathbb{D}) . We finally study the case where \partial \varphi (\mathbb{D}) meets the unit circle in one point and give several concrete examples. Our method is based on upper and lower estimates of the trace of h(T_\mu) , where h is a suitable concave or convex function.

The adjoint of the classic composition operator on the Hardy space of the unit disc determined by a holomorphic self map of the unit disc is well known to send the Szegő kernel function associated to a point in the unit disc to the Szegő kernel associated to the image of that point under the self map. The purpose of this paper is to show that a constructive proof that holomorphic functions that extend past the boundary can be well approximated by complex linear combinations of the Szegő kernel function gives an explicit formula for the adjoint of a composition operator that yields a new way of looking at these objects and provides inspiration for new ways of thinking about operators that act on linear spans of the Szegő kernel. Composition operators associated to multivalued self mappings will arise naturally, and out of necessity. A parallel set of ideas will be applied to composition operators on the Bergman space.

In this paper we study bounded and compact weighted composition operator, induced by a fixed analytic function and an analytic self-map of the open unit disk, from Bergman space into weighted Bloch space. As a corollary, obtain the characterization of composition operator from Bergman space into weighted Bloch space.

Let L2 = L2(D, r dr dθ/π) be the Lebesgue space on the open unit disc and let be the Bergman space. Let P be the orthogonal projection of L2 onto and let Q be the orthogonal projection onto . Then I − P ≥ Q. The big Hankel operator and the small Hankel operator on are defined as: for ϕ in L∞, and . In this paper, the finite‐rank intermediate Hankel operators between and are studied. We are working on the more general space, that is, the weighted Bergman space.

In this paper, we obtain a complete characterization for compact double differences formed by general composition operators on Bergman spaces over the half-plane. As applications, we notice that the famous Moorhouse cancellation property of compact double differences induced by three composition operators cannot occur in our setting. The relationship between angular derivatives and compact double differences of composition operators is also explored, which reveals some new phenomena due to the unboundedness of half-plane.

In this paper we consider linear operators M ψ C ϕ D and M ψ DC ϕ acting between weighted Bergman spaces, where M ψ , C ϕ and D are multiplication, composition and differentation operators respectively.Our goal is to characterize those holomorphic self maps ϕ of D for which M ψ DC ϕ and M ψ C ϕ D acts boundedly and compactly between weighted Bergman spaces.

For a wide class of radial weights we calculate the essential norm of a weighted composition operator \({uC_\varphi}\) on the weighted Banach spaces of analytic functions in terms of the analytic function \({u \colon \mathbb D \to \mathbb C}\) and the nth power of the analytic selfmap \({\varphi}\) of the open unit disc \({\mathbb D}\) . We also apply our result to calculate the essential norm of composition operators acting on Bloch type spaces with general radial weights.

Joint Carleson measure and the difference of composition operators on [formula omitted

We introduce the scale of weighted Bergman spaces on hyperconvex domains in C n and use the Lelong-Jensen formula to prove some fundamental results about these spaces. In particular, generalizations of such classical results as the Littlewood subordination principle, the Littlewood-Paley identity and the change of variables formula are proven. Geometric properties of the introduced norms are revealed by the Nevanlinna counting function associated with a chosen exhaustion. In the last several sections we prove boundedness and compactness results for composition operators generated by holomorphic mappings of hyperconvex domains.

A variety of norm inequalities related to Bergman and Dirichlet spaces induced by radial weights is established. Some of the results obtained can be considered as generalizations of certain known special cases while most of the estimates discovered are completely new. In particular, a Littlewood-Paley estimate recently proved by Peláez and the second author (Peláez and Rättyä Adv. Math. , 391 , 70, 2021) is improved in part. The second objective of the paper is to apply the obtained norm inequalities to relate the growth of the maximum modulus of a conformal map f , measured in terms of a weighted integrability condition, to a geometric quantity involving the area of image under f of a disc centered at the origin. Our findings in this direction yield new geometric characterizations of conformal maps in certain weighted Dirichlet and Besov spaces.

Chapter 39 Spaces of analytic functions with integral norm

Comparison of singular numbers of composition operators on different Hilbert spaces of analytic functions