Bounded composition operators with closed range on the Dirichlet space

For composition operators on spaces of analytic functions it is well known that norm estimates can be converted to Carleson measure estimates. The boundedness of the composition operator becomes equivalent to a Carleson measure inequality. The measure corresponding to a composition operator on the Dirichet space is , where is the cardinality of the preimage . The composition operator will have closed range if and only if the corresponding measure satisfies a ``reverse Carleson measure' theorem: for all . Assuming is bounded, a necessary condition for this inequality is a reverse of the Carleson condition: (C) for all Carleson squares . It has long been known that this is not sufficient for a completely general measure. Here we show that it is also not sufficient for the special measures . That is, we construct a function such that is bounded and satisfies (C) but the composition operator does not have closed range.

     

Composition operators on the polydisc induced by smooth symbols

We study composition operators CΦ on the Hardy spaces Hp and weighted Bergman spaces of the polydisc Dn in Cn. When Φ is of class C² on , we show that CΦ is bounded on Hp or if and only if the Jacobian of Φ does not vanish on those points ζ on the distinguished boundary Tn such that Φ(ζ)∈Tn. Moreover, we show that if ε>0 and if , then CΦ is bounded on .     

Extreme points of the closed convex hull of composition operators

We study the extreme points of the closed convex hull of the set of all composition operators on the space of bounded analytic functions and the disk algebra.     

Weighted holomorphic spaces with trivial closed range multiplication operators

We deal with the space consisting of those analytic functions on the unit disc such that , with . We determine the critical rate of decay of such that the pointwise multiplication operator , and analytic, has closed range in only in the trivial case that is the product of an invertible function in and a finite Blaschke product.

     

Adjoints of rationally induced composition operators

We give an elementary proof of a formula recently obtained by Hammond, Moorhouse, and Robbins for the adjoint of a rationally induced composition operator on the Hardy space H² [Christopher Hammond, Jennifer Moorhouse, Marian E. Robbins, Adjoints of composition operators with rational symbol, J. Math. Anal. Appl. 341 (2008) 626-639]. We discuss some variants and implications of this formula, and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator.     

Invertibility of linear differential operators with closed range and poisson coefficients

The invertibility and injectivity properties of linear differential operators with closed range and Poisson coefficients are studied in the context of their equivalence in several spaces of vector functions defined on the axis. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 143–147, January, 1999.     

Fredholm composition operators on algebras of analytic functions on Banach spaces

We prove that Fredholm composition operators acting on the uniform algebra H^∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.     

Some properties of unbounded operators with closed range

Let H ₁, H ₂ be Hilbert spaces and T be a closed linear operator defined on a dense subspace D(T) in H ₁ and taking values in H ₂. In this article we prove the following results:

A new class of operators and a description of adjoints of composition operators

Starting with a general formula, precise but difficult to use, for the adjoint of a composition operator on a functional Hilbert space, we compute an explicit formula on the classical Hardy Hilbert space for the adjoint of a composition operator with rational symbol. To provide a foundation for this formula, we study an extension to the definitions of composition, weighted composition, and Toeplitz operators to include symbols that are multiple-valued functions. These definitions can be made on any Banach space of analytic functions on a plane domain, but in this work, our attention is focused on the basic properties needed for the application to operators on the standard Hardy and Bergman Hilbert spaces on the unit disk.     

Norms of composition operators with rational symbol

We compute the norms of composition operators with rational symbols that satisfy certain properties, extending Christopher Hammond's methods on operators with linear fractional symbols. This leads to a host of new examples of composition operators whose norms are calculable.     

Adjoints of composition operators with rational symbol

Building on techniques developed by Cowen and Gallardo-Gutiérrez, we find a concrete formula for the adjoint of a composition operator with rational symbol acting on the Hardy space H². We consider some specific examples, comparing our formula with several results that were previously known.     

Products of integral-type operators and composition operators between Bloch-type spaces

A complete picture on the boundedness and compactness of the products of integral-type operators and composition operators between Bloch-type spaces of holomorphic functions on the unit disk is given in this paper.     

Toeplitz operators with unbounded symbols of several complex variables

In this note we construct a function φ in L²(Bn,dA) which is unbounded on any neighborhood of each boundary point of Bn such that Tφ is a trace class operator on Bergman space for several complex variables. In addition, we also discuss the compactness of Toeplitz operators with L¹ symbols.     

Adjoints of a class of composition operators

Adjoints of certain operators of composition type are calculated. Specifically, on the classical Hardy space of the open unit disk operators of the form are considered, where is a finite Blaschke product. is obtained as a finite linear combination of operators of the form where and are rational functions, are associated Toeplitz operators and is defined by

     

Separating classes of composition operators via subnormal condition

Several classes have been considered to study the weak subnormalities of Hilbert space operators. One of them is -hypnormality, which comes from the Bram-Halmos criterion for subnormal operators. In this note we consider -hyponormality, which is the parallel version corresponding to the Embry characterization for subnormal operators. We characterize -hyponormality of composition operators via -th Radon-Nikodym derivatives and present some examples to distinguish the classes.

     

On the Toeplitzness of the adjoint of composition operators

Building on techniques developed by Cowen (1988)  [3] and Nazarov–Shapiro (2007)  [10], it is shown that the adjoint of a composition operator, induced by a unit disk-automorphism, is not strongly asymptotically Toeplitz. This result answers Nazarov–Shapiro’s question in Nazarov and Shapiro (2007)  [10].     

Compact differences of composition operators

A characterization of compact difference is given for composition operators acting on the standard weighted Bergman spaces and necessary conditions are given on a larger scale of weighted Dirichlet spaces. Conditions are given under which a composition operator can be written as a finite sum of composition operators modulo the compacts. The additive structure of the space of composition operators modulo the compact operators is investigated further and a sufficient condition is given to insure that two composition operators lie in the same component.     

Commuting Toeplitz operators with pluriharmonic symbols

By making use of -harmonic function theory, we characterize commuting Toeplitz operators with bounded pluriharmonic symbols on the Bergman space of the unit ball or on the Hardy space of the unit sphere in -dimensional complex space.