Essential norm estimates of weighted composition operators between Bergman spaces on strongly pseudoconvex domains

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.

Norm Approximation by Polynomials in Some Weighted Bergman Spaces

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 \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>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.

We characterize bounded and compact weighted composition operators acting between weighted Bergman spaces and between Hardy spaces. Our results use certain integral transforms that generalize the Berezin transform. We also estimate the essential norms of these operators. As applications, we characterize bounded and compact pointwise multiplication operators between weighted Bergman spaces and estimate their essential norms.

Consider an open unit disk D = z ∈ ℂ : z < 1 in the complex plane ℂ , ξ a holomorphic function on D , and ψ a holomorphic self-map of D . For an analytic function f , the weighted composition operator is denoted and defined as follows: W ξ , ψ f z = ξ z f ψ z . We estimate the essential norm of this operator from Dirichlet-type spaces to Bers-type spaces and Bloch-type spaces.

In the paper we give a lower estimate for the essential norm of the $\bar{\partial}$-Neumann operator on convex domains and worm domains of Diederich and Fornaess.

We give a new characterization for the boundedness of composi- tion operator followed by dierentiation operator acting on Bloch-type spaces and calculate its essential norm in terms of the n-th power of the induced an- alytic self-map on the unit disk. From which some sucient and necessary conditions of compactness of the operator follow immediately.

Let D be a bounded strongly pseudoconvex domain with smooth boundary in Cn and let f∈L2(D). For the Hankel operator Hf on the Bergman space A2(D), it is shown that the essential norm of Hf in L2(D) is comparable to the distance norm from Hf to compact Hankel operators. The result extends the previous corresponding version in the disc proved by Lin and Rochberg in Integ.Equat.Oper.Theory 361–372,17 (1993).

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.

Let A p α (B n ; C d ) be the weighted Bergman space on the unit ball B n of C n of functions taking values in C d .For 1 < p < ∞ let T p,α be the algebra generated by finite sums of finite products of Toeplitz operators with bounded matrix-valued symbols (this is called the Toeplitz algebra in the case d = 1).We show that every S ∈ T p,α can be approximated by localized operators.This will be used to obtain several equivalent expressions for the essential norm of operators in T p,α .We then use this to characterize compact operators in A p α (B n ; C d ).The main result generalizes previous results and states that an operator in A p α (B n ; C d ) is compact if only if it is in T p,α and its Berezin transform vanishes on the boundary.

We consider a fairly large class of Hilbert spaces of holomorphic functions in one complex variable which contains the Hardy, the Fock and the Bergman space. The purpose of our paper is to study linear bounded operators on these spaces by means of their matrix elements (Tek,ej) where ek are the normalized monomials. To this end we introduce a “Fourier series expansion” of T into special operators and discuss how the properties of T are reflected in its “Fourier summands”. We characterize, among others, the essential spectrum and the essential norm for a large class of T including many Toeplitz operators.

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.

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^2$ with Lipschitz boundary or a bounded convex domain in $\mathbb{C}^n$ and $\phi\in C(\overline{\Omega})$ such that $H_{\phi}$ is compact on $A^2(\Omega)$. Then $\phi\circ f$ is holomorphic for any holomorphic $f:\mathbb{D}\rightarrow b\Omega$.