Composition operators on Hardy spaces of a half-plane

We consider composition operators on Hardy spaces of a half-plane. We mainly study boundedness and compactness. We prove that on these spaces there are no compact composition operators.

     

Components of linear-fractional composition operators

We determine when two linear-fractional composition operators on the Hardy space H² belong to the same component in the collection of all composition operators on H². We show that two such composition operators in the same component may fail to have compact difference, which answers a question raised by Joel Shapiro and Carl Sundberg.     

Composition operators on weighted Dirichlet spaces

We characterize bounded and compact composition operators on weighted Dirichlet spaces. The method involves integral averages of the determining function for the operator, and the connection between composition operators on Dirichlet spaces and Toeplitz operators on Bergman spaces. We also present several examples and counter-examples that point out the borderlines of the result and its connections to other themes.

     

Which linear-fractional composition operators are essentially normal?

We characterize the essentially normal composition operators induced on the Hardy space H² by linear-fractional maps; they are either compact, normal, or (the nontrivial case) induced by parabolic nonautomorphisms. These parabolic maps induce the first known examples of nontrivially essentially normal composition operators. In addition, we characterize those linear-fractionally induced composition operators on H² that are essentially self-adjoint, and present a number of results for composition operators induced by maps that are not linear-fractional.     

Aluthge transforms of unbounded weighted composition operators in L2-spaces

We describe the Aluthge transform of an unbounded weighted composition operator acting in an L²-space. We show that its closure is also a weighted composition operator with the same symbol and a modified weight function. We investigate its dense definiteness. We characterize p-hyponormality of unbounded weighted composition operators and provide results on how it is affected by the Aluthge transformation. We show that the only fixed points of the Aluthge transformation on weighted composition operators are quasinormal ones.     

Carleson embeddings and some classes of operators on weighted Bergman spaces

We prove Carleson-type embedding theorems for weighted Bergman spaces with Békollé weights. We use this to study properties of Toeplitz-type operators, integration operators and composition operators acting on such spaces. In particular, we investigate the membership of these operators to Schatten class ideals.     

Quasinormal and Hyponormal Weighted Composition Operators on $$H^2$$ and $$A^2_{\alpha }$$Aα2 with Linear Fractional Compositional Symbol

In this paper, we study quasinormal and hyponormal composition operators \(W_{\psi ,\varphi }\)  with linear fractional compositional symbol \(\varphi \) on the Hardy and weighted Bergman spaces. We characterize the quasinormal composition operators induced on \(H^{2}\) and \(A_{\alpha }^{2}\) by these maps and many such weighted composition operators, showing that they are necessarily normal in all known cases. We eliminate several possibilities for hyponormal weighted composition operators but also give new examples of hyponormal weighted composition operators on \(H^2\) which are not quasinormal.     

Topological Structure of the Set of Weighted Composition Operators on Weighted Bergman Spaces of Infinite Order

We consider the topological space of all weighted composition operators on weighted Bergman spaces of infinite order endowed with the operator norm. We show that the set of compact weighted composition operators is path connected. Furthermore, we find conditions to ensure that two weighted composition operators are in the same path connected component if the difference of them is compact. Moreover, we compare the topologies induced by L(H^∞) and L(H^∞_v) on the space of bounded composition operators and give a sufficient condition for a composition operator to be isolated.     

Joint hyponormality of composition operators with linear fractional symbols

We study joint hyponormality and joint subnormality of ofn-tuples of commuting composition operators with linear fractional symbols, acting on the Hardy spaceH ². We also consider subnormality ofn-tuples of adjoints of composition operators.     

Generalized composition and Volterra type operators between QK spaces

Abstract

In this paper, the boundedness and compactness of the generalized composition operators and the products of Volterra type operators a nd composition operators between Q_K spaces are investigated. We also give a necessary condition for multiplication operators between Q_K spaces to be bounded or compact.     

Composition operators between Bergman and Hardy spaces

We study composition operators between weighted Bergman spaces. Certain growth conditions for generalized Nevanlinna counting functions of the inducing map are shown to be necessary and sufficient for such operators to be bounded or compact. Particular choices for the weights yield results on composition operators between the classical unweighted Bergman and Hardy spaces.

     

Weighted Composition Operators on H 2 and Applications

Operators on function spaces acting by composition to the right with a fixed selfmap φ of some set are called composition operators of symbol φ. A weighted composition operator is an operator equal to a composition operator followed by a multiplication operator. We summarize the basic properties of bounded and compact weighted composition operators on the Hilbert Hardy space on the open unit disk and use them to study composition operators on Hardy–Smirnov spaces. Submitted: January 30, 2007. Revised: June 19, 2007. Accepted: July 11, 2007.     

Norm-attaining weighted composition operators on weighted Banach spaces of analytic functions

We investigate weighted composition operators that attain their norm on weighted Banach spaces of holomorphic functions on the unit disc of type H ^∞. Applications for composition operators on weighted Bloch spaces are given.     

Composition operators acting on holomorphic Sobolev spaces

We study the action of composition operators on Sobolev spaces of analytic functions having fractional derivatives in some weighted Bergman space or Hardy space on the unit disk. Criteria for when such operators are bounded or compact are given. In particular, we find the precise range of orders of fractional derivatives for which all composition operators are bounded on such spaces. Sharp results about boundedness and compactness of a composition operator are also given when the inducing map is polygonal.

     

Differences of Weighted Composition Operators on the Bloch Spaces

We study the boundedness and the compactness of the differences of weighted composition operators on the Bloch spaces. The results generalize the corresponding results on the single weighted composition operators and on the differences of composition operators.     

Linear Combinations of Composition Operators on the Fock-Sobolev Spaces

We study linear combinations of composition operators acting on the Fock-Sobolev spaces of several variables. We show that such an operator is bounded only when all the composition operators in the combination are bounded individually. In other words, composition operators on the Fock-Sobolev spaces do not possess the same cancelation properties as composition operators on other well-known function spaces over the unit disk. We also show the analogues for compactness and the membership in the Schatten classes. In particular, compactness and the membership in some/all of the Schatten classes turn out to be the same.     

Variational principle for the spectral exponent of polynomials of weighted composition operators

In this paper we derive a relationship between the Legendre-Fenchel transform of the spectral exponent of weighted composition operators acting in Lp-spaces and the Legendre-Fenchel transform obtained for their polynomials. We establish the variational principle for the spectral exponent of polynomials of weighted composition operators.     

Generalized Essential Norm of Weighted Composition Operators on some Uniform Algebras of Analytic Functions

We compute the essential norm of a composition operator relatively to the class of Dunford-Pettis operators or weakly compact operators, on some uniform algebras of analytic functions. Even in the context of H^∞ (resp. the disk algebra), this is new, as well for the polydisk algebras and the polyball algebras. This is a consequence of a general study of weighted composition operators.      

Closed-Range Composition Operators on Dirichlet-Type Spaces

We characterize the bounded composition operators mapping one Dirichlet-type space into another. We also give a geometric characterization for those composition operators having closed range.     

Schur spaces and weighted spaces of type H ∞

Abstract

We extend some results related to composition operators on H _υ(G) to arbitrary linear operators on H _υ0(G) and H _υ(G). We also give examples of rank-one operators on H _υ(G) which cannot be approximated by composition operators.