Essential normality of polynomial-generated submodules: Hardy space and beyond

Recently, Douglas and Wang proved that for each polynomial q  , the submodule [q]$[q]$ of the Bergman module generated by q is essentially normal [11]. Using improved techniques, we show that the Hardy space analogue of this result holds, and more.     

Fredholm multiplication and composition operators on the Hardy space

We present a unified approach to the problem of characterizing the Fredholm multiplication and composition operators on the Hardy spaceH ².     

Some thoughts on composition operators on subspaces of the Hardy space

Archiv der Mathematik - We discuss composition operators on certain subspaces of the Hardy space. The family of subspaces that we deal with are called $$H^2_{alpha , beta }$$ which have garnered...     

Essential norm of composition operators between weighted Hardy spaces

An asymptotic formula for the essential norm of composition operators acting between two weighted Hardy spaces H_w1 and H_w2, where w₁ and w₂ are two admissible weight functions, is given. The boundedness of the operators is also characterized.     

On the connected component of compact composition operators on the Hardy space

We show that there exist non-compact composition operators in the connected component of the compact ones on the classical Hardy space H². This answers a question posed by Shapiro and Sundberg in 1990. We also establish an improved version of a theorem of MacCluer, giving a lower bound for the essential norm of a difference of composition operators in terms of the angular derivatives of their symbols. As a main tool we use Aleksandrov-Clark measures.     

Complete continuity of linear combinations of composition operators

In this note, we characterize complete continuity of linear combinations of composition operators on Hardy and Bergman spaces of the unit disk.     

Disjoint mixing composition operators on the Hardy space in the unit ball

We characterize disjoint mixing and disjoint hypercyclicity of finite many composition operators acting on the Hardy space on the unit ball.     

Singular values and Schmidt pairs of composition operators on the Hardy space

We find the singular values and corresponding Schmidt pairs of a compact composition operator Cφ induced by φ(z)=az+b, where |a|+|b|<1, on the classical Hardy space. We do so by solving a functional equation that is a generalization of Schröder's equation: find a function f, holomorphic on the open unit disc, and a complex number λ such that G(z)f(ψ(z))=λf(ψ(z)), where ψ is a holomorphic self-map of the open unit disc with an interior fixed point and G is a bounded holomorphic function on the open unit disc. In addition, we find the spectrum of the weighted composition operator MGCψ.     

Essential spectra of quasi-parabolic composition operators on Hardy spaces of analytic functions

In this work we study the essential spectra of composition operators on Hardy spaces of analytic functions which might be termed as “quasi-parabolic.” This is the class of composition operators on H² with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form φ(z)=z+ψ(z), where ψ∈H^∞(H) and ℑ(ψ(z))>?>0. We especially examine the case where ψ is discontinuous at infinity. A new method is devised to show that this type of composition operator fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.     

Widths of certain classes of analytic functions in the Hardy space H2

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 6, pp. 799–803, June, 1989.     

Adjoints of linear fractional composition operators on the Dirichlet space

The adjoint of a linear fractional composition operator acting on the classical Dirichlet space is expressed as another linear fractional composition operator plus a two rank operator. The key point is that, in the Dirichlet space modulo constant functions, many linear fractional composition operators are similar to multiplication operators and, thus, normal. As a particular application, we can easily deduce the spectrum of each linear fractional composition operator acting on such spaces. Even the norm of each linear fractional composition operator is computed on the Dirichlet space modulo constant functions. It is also shown that all this work can be carried out in the Hardy space of the upper half plane.This work was partially supported by Plan Nacional I+D Ref. BFM2000-0360 and Junta de Andalucía Ref. FQM-260. The first named author was also supported by Plan Propio de la Universidad de Cádiz.     

Essential norm of the difference of composition operators on Bloch space

Let ϕ and ψ be holomorphic self-maps of the unit disk, and denote by C _ϕ , C _ψ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators C _ϕ −C _ψ from Bloch spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized.     

Weighted composition operators from the weighted Bergman space to the weighted Hardy space on the unit ball

We investigate the weighted composition operator from the weighted Bergman space into the weighted Hardy space on the unit ball. As a consequence of the investigation, we also give a characterization for the boundedness and compactness of the operator whose the target space is the Hardy space.     

Essential norm of composition operators on the Hardy space H 1 and the weighted Bergman spaces {A_{\alpha}^{p}} on the ball

We estimate the essential norm of a composition operator acting on the Hardy space H ¹ and the weighted Bergman spaces ${A_{\alpha}^{p}}$ on the unit ball. In passing, we recover (and somehow simplify the proof of) parts of the recent article by Demazeux, dealing with the same question for H ¹ of the unit disc. We also estimate the essential norm of a composition operator acting on ${A_{\alpha}^{p}}$ in terms of the angular derivatives of ${\phi}$ , under a mild condition on ${\phi}$ .     

Essential norm and a new characterization of weighted composition operators from weighted Bergman spaces and Hardy spaces into the Bloch space

In this paper, we give some estimates for the essential norm and a new characterization for the boundedness and compactness of weighted composition operators from weighted Bergman spaces and Hardy spaces to the Bloch space.     

Commuting Toeplitz operators on the Hardy space of the bidisk

On the Hardy space of the bidisk, we give a necessary and sufficient condition for a bounded symbol of a Toeplitz operator that commutes with another Toeplitz operator whose symbol is a certain type of bounded symbol.     

Norms of composition operators on weighted Hardy spaces

The norm of a bounded composition operator induced by a disc automorphism is estimated on weighted Hardy spaces H ²(β) in which the classical Hardy space is continuously embedded. The estimate obtained is accurate in the sense that it provides the exact norm for particular instances of the sequence β. As a by-product of our results, an estimate for the norm of any bounded composition operator on H ²(β) is obtained.     

Essential norms of weighted composition operators from weighted Bergman space to mixed-norm space on the unit ball

In this paper, we characterize the boundedness and compactness of the weighted composition operators from the weighted Bergman space to the standard mixed-norm space or the mixed-norm space with normal weight on the unit ball and estimate the essential norms of the weighted composition operators.