Composition operators on Hardy spaces of a half-plane

It is shown that there exist analytic self-maps ϕ of the unit disc inducing compact composition operators on the Hardy space , 1 ≤ p < ∞ such that the Hausdorff dimension of the set is one; sharpening a classical result due to Schwartz. Moreover, the same holds in the weighted Dirichlet spaces with 0 < α < 1. As a consequence, we deduce that there exist symbols ϕ inducing compact composition operators on such that the α-capacity of E ϕ is positive, which is no longer true for those just inducing Hilbert-Schmidt composition operators on .

We generalise previous results of the author concerning the compactness of composition operators on the Hardy spaces $H^{p}$, $1\leq p<\infty$, whose symbol is a universal covering map from the unit disk in the complex plane to general finitely connected domains. We demonstrate that the angular derivative criterion for univalent symbols extends to this more general case. We further show that compactness in this setting is equivalent to compactness of the composition operator induced by a univalent mapping onto the interior of the outer boundary component of the multiply connected domain.

We show that a restriction on the angular derivative of the inducing map does not determine compact composition operators on large weighted Hardy spaces, thus answering in the negative a question posed by T. Kriete.

We give a new and simple compactness criterion for composition operators C ϕ on BMOA and the Bloch space in terms of the norms of ϕ n in the respective spaces.

The composition operator induced by a holomorphic self-map of the unit disc is compact on L 1 {L^1} of the unit circle if and only if it is compact on the Hardy space H 2 {H^2} of the disc. This answers a question posed by Donald Sarason: it proves that Sarason’s integral condition characterizing compactness on L 1 {L^1} is equivalent to the asymptotic condition on the Nevanlinna counting function which characterizes compactness on H 2 {H^2} .

Compact composition operators on generalized Lipschitz spaces

In this paper, we study a composition operator \({C_{\varphi}}\) on the weighted Bergman space \({A_{\alpha}^p(B)}\) of the unit ball B in \({{\mathbb{C}}^N}\) . Under a natural condition we estimate the essential norm of \({C_{\varphi}}\) . As a consequence of this estimate, we also give a function-theoretic characterization of \({\varphi}\) that induces a compact composition operator on \({A_{\alpha}^p(B)}\) .

Composition Operators from Bloch Type Spaces to Hardy and Besov Spaces

We give embedding theorems for Hardy–Orlicz spaces on the ball and then apply our results to the study of the boundedness and compactness of composition operators in this context. As one of the motivations of this work, we show that there exist some Hardy–Orlicz spaces, different from H ∞, on which every composition operator is bounded.

We characterize the closure of Dirichlet type spaces Dμ in Bloch type spaces. Moreover, we study the boundedness and compactness of composition operators from Bloch type spaces to closures of Dirichlet type spaces Dμ in Bloch type spaces.

We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space and the analytic Besov spaces (with ) into the Zygmund space . As a special case, we show that the bounded (resp., compact) composition operators from , , and to coincide. In addition, the boundedness and the compactness of the composition operator can be characterized in terms of the boundedness (resp., convergence to 0, under the boundedness assumption of the operator) of the Zygmund norm of the powers of the symbol.

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL α 1 (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range.

Weakly compact composition operators on vector-valued BMOA

We give a Carleson measure characterization of the compact composition operators on Besov spaces. We use this characterization to show that every compact composition operator on a Besov space is compact on the Bloch space. Finally we give conditions that guarantee that the converse holds.

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.