Given a holomorphic self-map $\varphi $ of $\mathbb {D}$ (the open unit disc in $\mathbb {C}$ ), the composition operator $C_{\varphi } f = f \circ \varphi $ , $f \in H^2(\mathbb {\mathbb {D}})$ , defines a bounded linear operator on the Hardy space $H^2(\mathbb {\mathbb {D}})$ . The model spaces are the backward shift-invariant closed subspaces of $H^2(\mathbb {\mathbb {D}})$ , which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.

For an inner function u, we discuss the dual operator for the compressed shift $P_u S|_{{\mathcal {K}}_u}$, where ${\mathcal {K}}_u$ is the model space for u. We describe the unitary equivalence/similarity classes for these duals as well as their invariant subspaces.

We study the image of the model subspace ${{K}_{\theta }}$ under the composition operator ${{C}_{\varphi }}$, where $\varphi $ and $\theta $ are inner functions, and find the smallest model subspace which contains the linear manifold ${{C}_{\varphi }}{{K}_{\theta }}$. Then we characterize the case when ${{C}_{\varphi }}$ maps ${{K}_{\theta }}$ into itself. This case leads to the study of the inner functions $\varphi $ and $\psi $ such that the composition $\psi \,\text{o}\,\varphi $ is a divisor of $\psi $ in the family of inner functions.

We give derivative-free characterizations for bounded and compact generalized composition operators between (little) Zygmund type spaces. To obtain these results, we extend Pavlović’s corresponding result for bounded composition operators between analytic Lipschitz spaces.

For a function u harmonic in the unit disk , there holds the inequality where p > 0 and β > −1, and .

We study the membership of derivatives of Blaschke products in Hardy and Bergman spaces, especially for the the interpolating Blaschke products and for those whose zeros lie in a Stolz domain. We obtain new and very simple proofs of some known results and prove new theorems that complement or extend the earlier works of Ahern, Clark, Cohn, Kim, Newman, Protas, Rudin, Vinogradov and others.

Sufficient conditions for the analytic coefficients of the linear differential equation

are found such that all solutions belong to a given -space, or to the Dirichlet type subspace Dp of the classical Hardy space Hp, where 0 < p ≤ 2. For 0 < q < ∞, the space consists of those functions f, analytic in the unit disc D, for which |f(z)|(1 – |z|2)q is uniformly bounded in D, and f ∈ Dp if the integral ∫D |f′(z)|p(1 – |z|2)p–1dσz converges.

It is known that the derivative of a Blaschke product whose zero sequence lies in a Stolz angle belongs to all the Bergman spaces ${{A}^{P}}$ with $0<p<3/2$. The question of whether this result is best possible remained open. In this paper, for a large class of Blaschke products $B$ with zeros in a Stolz angle, we obtain a number of conditions which are equivalent to the membership of ${B}'$ in the space ${{A}^{p}}\left( p>1 \right)$. As a consequence, we prove that there exists a Blaschke product $B$ with zeros on a radius such that ${B}'\,\notin \,{{A}^{3/2}}$.

For 0 < p < ∞, we let pp−1 denote the space of those functions f that are analytic in the unit disc Δ = {z ∈ C: |z| < 1} and satisfy ∫Δ(1 – |z|)p−1|f′(z)|pdx dy < ∞ The spaces pp−1 are closely related to Hardy spaces. We have, p−1p ⊂ Hp, if 0 < p ≦ 2, and Hp ⊂ pp−1, if 2 ≦ p < ∞. In this paper we obtain a number of results about the Taylor coefficients of pp-1 -functions and sharp estimates on the growth of the integral means and the radial growth of these functions as well as information on their zero sets.

2000 Mathematics subject classification: primary 30D35, 30D55, 46E15.

Riemann–Stieltjes integrals are considered as linear operators on weighted Bloch and Bergman spaces of the open unit ball in several complex variables. For weighted Bloch spaces, boundedness, compactness and weak compactness of Riemann–Stieltjes operators are characterized by means of certain growth properties of holomorphic symbols. For weighted Bergman spaces, some criteria are given for Riemann–Stieltjes operators with holomorphic symbols to be bounded, compact and of Schatten–von Neumann's ideal.

A model subspace ${{K}_{\Theta }}$ of the Hardy space ${{H}^{2}}={{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)$ for the upper half plane ${{\mathbb{C}}_{+}}$ is ${{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)\ominus \Theta {{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)$ where $\Theta $ is an inner function in ${{\mathbb{C}}_{+}}$. A function $\omega :\,\mathbb{R}\mapsto [0,\,\infty )$ is called an admissible majorant for ${{K}_{\Theta }}$ if there exists an $f\,\in \,{{K}_{\Theta }},\,f\,\not{\equiv }\,0,\,|f\left( x \right)|\,\le \,\omega \left( x \right)$ almost everywhere on $\mathbb{R}$. For some (mainly meromorphic) $\Theta $'s some parts of Adm $\Theta $ (the set of all admissible majorants for ${{K}_{\Theta }}$) are explicitly described. These descriptions depend on the rate of growth of arg $\Theta $ along $\mathbb{R}$. This paper is about slowly growing arguments (slower than $x$). Our results exhibit the dependence of Adm $B$ on the geometry of the zeros of the Blaschke product $B$. A complete description of Adm $B$ is obtained for $B$'s with purely imaginary (“vertical”) zeros. We show that in this case a unique minimal admissible majorant exists.

This paper is a continuation of $[6]$. We consider the model subspaces ${{K}_{\Theta }}={{H}^{2}}\ominus \Theta {{H}^{2}}$ of the Hardy space ${{H}^{2}}$ generated by an inner function $\Theta $ in the upper half plane. Our main object is the class of admissible majorants for ${{K}_{\Theta }}$, denoted by Adm $\Theta $ and consisting of all functions $\omega $ defined on $\mathbb{R}$ such that there exists an $f\ne 0,f\in {{K}_{\Theta }}$ satisfying $|f\left( x \right)|\,\le \,\omega \left( x \right)$ almost everywhere on $\mathbb{R}$. Firstly, using some simple Hilbert transform techniques, we obtain a general multiplier theorem applicable to any ${{K}_{\Theta }}$ generated by a meromorphic inner function. In contrast with $[6]$, we consider the generating functions $\Theta $ such that the unit vector $\Theta \left( x \right)$ winds up fast as $x$ grows from $-\infty \,\text{to}\,\infty $. In particular, we consider $\Theta \,=\,B$ where $B$ is a Blaschke product with “horizontal” zeros, i.e., almost uniformly distributed in a strip parallel to and separated from $\mathbb{R}$. It is shown, among other things, that for any such $B$, any even $\omega $ decreasing on $\left( 0,\,\infty \right)$ with a finite logarithmic integral is in Adm $B$ (unlike the “vertical” case treated in $[6]$), thus generalizing (with a new proof) a classical result related to Adm $\exp \left( i\sigma z \right),\,\sigma \,>\,0$. Some oscillating $\omega $'s in Adm $B$ are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm $\exp \left( i\sigma z \right),\,\sigma \,>\,0$, and to de Branges’ space $H\left( E \right)$.

Every classical inner function $\varphi$ in the unit disk gives rise to a certain factorization of functions in Hardy spaces. This factorization, which we call the generalized Riesz factorization, coincides with the classical Riesz factorization when $\varphi (z)\,=\,z$. In this paper we prove several results about the generalized Riesz factorization, and we apply this factorization theory to obtain a new description of the commutant of analytic Toeplitz operators with inner symbols on a Hardy space. We also discuss several related issues in the context of the Bergman space.

On the half space Rn × R+, it has been known that harmonic Bergman space bp can contain a positive function only if . Thus, for , Poisson integrals can be bp-functions only by means of their boundary cancellation properties. In this paper, we describe what those cancellation properties explicitly are. Also, given such cancellation properties, we obtain weighted norm inequalities for Poisson integrals. As a consequence, under weighted integrability condition given by our weighted norm inequalities, we show that our cancellation properties are equivalent to the bp-containment of Poisson integrals for p under consideration. Our results are sharp in the sense that orders of our weights cannot be improved.

Every weakly compact composition operator between weighted Banach spaces $H_{v}^{\infty }$ of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space $H_{v}^{\infty }$ are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

Normal functions and Bloch functions are respectively functions of bounded spherical expansion and bounded Euclidean expansion. In this paper we discuss the behaviour of normal functions and of Bloch functions in terms of the maximal ideal space of H∞, the Bergman projection and the Ahlfors-Shimizu characteristic.

In this paper we consider questions of recapturing an analytic function in a Banach space from its values on a uniqueness set. The principal method is to use reproducing kernels to construct a sequence in the Banach space which converges in norm to the given functions. The method works for several classical Banach spaces of analytic functions including some Hardy and Bergman spaces.