Interpolating Blaschke Products Research Articles

Interpolation and duality in spaces of pseudocontinuable functions

Given an inner function theta on the unit disk, let K^p_theta :=H^pcap theta {overline{z}}overline{H^p} be the associated star-invariant subspace of the Hardy space H^p. Also, we put K_{*theta }:=K^2_theta cap mathrm{BMO}. Assuming that B=B_{{mathcal {Z}}} is an interpolating Blaschke product with zeros {mathcal {Z}}={z_j}, we characterize, for a number of smoothness classes X, the sequences of values {mathcal {W}}={w_j} such that the interpolation problem fbig |_{{mathcal {Z}}}={mathcal {W}} has a solution f in K^2_Bcap X. Turning to the case of a general inner function theta , we further establish a non-duality relation between K^1_theta and K_{*theta }. Namely, we prove that the latter space is properly contained in the dual of the former, unless theta is a finite Blaschke product. From this we derive an amusing non-interpolation result for functions in K_{*B}, with B=B_{{mathcal {Z}}} as above.

Schur–Nevanlinna parameters, Riesz bases, and compact Hankel operators on the model space

We study Riesz bases/Riesz sequences of reproducing kernels in the model space Kθ in connection with the corresponding Schur–Nevanlinna parameters and functions. In particular, we construct inner functions with given Schur–Nevanlinna parameters at a given sequence Λ such that the corresponding systems of projections of reproducing kernels in the model space are complete/non-complete. Furthermore, we give a compactness criterion for Hankel operators with symbol θB‾, where θ is an inner function and B is an interpolating Blaschke product, and use this criterion to describe the Riesz bases KΛ,θ with limλ∈Λ,|λ|→1⁡θ(λ)=0.

A new characterization of the closure of Dirichlet type spaces Ds in Bloch spaces and interpolating Blaschke product

In this paper, motivated by Qian, et al [20, 22], we give a new characterization for the closure of the space Ds in the Bloch space.

Asymmetric Truncated Hankel Operators: Rank One, Matrix Representation

Asymmetric truncated Hankel operators are the natural generalization of truncated Hankel operators. In this paper, we determine all rank one operators of this class. We explore these operators on finite-dimensional model spaces, in particular, their matrix representation. We also give their matrix representation and the one for asymmetric truncated Toeplitz operators in the case of model spaces associated to interpolating Blaschke products.

Characterizations of Hankel operators in the essential commutant of quasicontinuous Toeplitz operators

AbstractThis note characterizes, in terms of interpolating Blaschke products, the symbols of Hankel operators essentially commuting with all quasicontinuous Toeplitz operators on the Hardy space of the unit circle. It also shows that such symbols do not contain the complex conjugate of any nonconstant singular inner function.

Multi-orbital Frames Through Model Spaces

We characterize the normal operators A on \(\ell ^2\) and the elements \(a^i \in \ell ^2\), with \(1\le i\le m\), such that the sequence $$\begin{aligned} \{ A^n a^1, \ldots , A^n a^m \}_{n\ge 0} \end{aligned}$$is a frame. The characterization makes strong use of the pseudo-hyperbolic metric of \( {{\mathbb {D}}} \) and is given in terms of the backward shift invariant subspaces of \(H^2( {{\mathbb {D}}} )\) associated to finite products of interpolating Blaschke products.

On Compact Perturbations of Hankel Operators and Commutators of Toeplitz and Hankel Operators

Motivated by results in uniform algebras, a distance localization formula in C*-algebras is established in the framework of the Allan–Douglas localization principle, and is used to derive a locality result for products of Hankel operators as compact perturbations of Hankel operators. Using localization and certain QC functions, it is proved that the essential spectrum of the commutator of a Toeplitz and a Hankel operator is antipodal symmetric under a mild condition on the Hankel symbol function. Under the same condition the essential spectrum of a Hankel operator also exhibits this symmetry. Conjugates of interpolating Blaschke products and characteristic functions are constructed that satisfy the condition, while examples show the condition is only sufficient.

Closed Range Type Properties of Toeplitz Operators on the Bergman Space and the Berezin Transform

We characterize the multiplication operators with closed range on the Bergman space in terms of the Berezin transform, and apply this characterization to finite products of interpolating Blaschke products. We give some necessary and some sufficient conditions for invertibility of general Toeplitz operators on the Bergman space. We determine the Fredholm Toeplitz operators with $$BMO^1$$ symbols and the invertible Toeplitz operators with nonnegative symbols, when their Berezin transform is bounded and of vanishing oscillation.

Interpolating by Functions from Model Subspaces in $$H^1$$ H 1

Given an interpolating Blaschke product $B$ with zeros $\{a_j\}$, we seek to characterize the sequences of values $\{w_j\}$ for which the interpolation problem $$f(a_j)=w_j\qquad (j=1,2,\dots)$$ can be solved with a function $f$ from the model subspace $H^1\cap B\overline{H^1_0}$ of the Hardy space $H^1$.

A free interpolation problem for a subspace of H∞

Given an inner function $\theta$, the associated star-invariant subspace $K^\infty_\theta$ is formed by the functions $f\in H^\infty$ that annihilate (with respect to the usual pairing) the shift-invariant subspace $\theta H^1$ of the Hardy space $H^1$. Assuming that $B$ is an interpolating Blaschke product with zeros $\{a_j\}$, we characterize the traces of functions from $K^\infty_B$ on the sequence $\{a_j\}$. The trace space that arises is, in general, non-ideal (i.e., the sequences $\{w_j\}$ belonging to it admit no nice description in terms of the size of $|w_j|$), but we do point out explicit -- and sharp -- size conditions on $|w_j|$ which make it possible to solve the interpolation problem $f(a_j)=w_j$ ($j=1,2,\dots$) with a function $f\in K^\infty_B$.

On the Closures of Dirichlet Type Spaces in the Bloch Space

In this paper, via high order derivatives and via embedding derivatives of Bloch type functions into Lebesgue spaces, we characterize the closures of Dirichlet type spaces in the Bloch space. We obtain the inclusion relation between the closures and the little Bloch space. We consider the separability of the closures as Banach spaces. A criterion for an interpolating Blaschke product to be in the closures is given. We also consider the relation between the closures and the space of bounded analytic functions.

Finitely generated radical ideals in the Sarason algebra

It is shown that a radical ideal in H∞+C is finitely generated if and only if it is the zero ideal or a principal ideal generated by an interpolating Blaschke product.

Interpolating Blaschke products and angular derivatives

We show that to each inner function, there corresponds at least one interpolating Blaschke product whose angular derivatives have precisely the same behavior as the given inner function. We characterize the Blaschke products invertible in the closed algebra H-infinity[(b) over bar : b has finite angular derivative everywhere]. We study the most well-known example of a Blaschke product with infinite angular derivative everywhere and show that it is an interpolating Blaschke product. We conclude the paper with a method for constructing thin Blaschke products with infinite angular derivative everywhere.

Quasi-Universal Functions for Sequences of Composition Operators on H 2 Via Hoffman’s Theory

Let \({\phi}\) be a non-elliptic automorphism of the unit disk \({\mathbb D}\). Gallardo and Gorkin (respectively Gallardo, Gorkin and Suarez) showed that there exists an interpolating Blaschke products b (respectively a thin Blaschke product) such that the linear span of its orbit under the composition operator \({C_\phi}\) is dense in H2 (in other words that b is a cyclic vector for \({C_\phi}\)). Using Hoffman’s theory, we extend their result from the automorphic case to the selfmap case and show that for any sequence \({(\phi_n)}\) of holomorphic selfmaps of \({\mathbb D}\) with \({|\phi_n(0)|\to 1}\) and $$\begin{array}{ll}\limsup\limits_{n\to\infty}\frac{|\phi_n'(0)|}{1-|\phi_n(0)|^2}=1,\end{array}$$ there exists a thin Blaschke product B with zero distribution as sparse as one wishes such that the linear span of the orbit \({\{B\circ \phi_n:n\in\mathbb N\}}\) of B is dense in H2. We will dub such a function a quasi-universal function. We also describe the limit points in the topological space \({M(H^\infty)^\mathbb D}\) of these admissible sequences \({(\phi_n)}\). This approach will enable us to give shorter proofs of the Gallardo–Gorkin result as well as the result of Bayart, Gorkin, Grivaux and the author on the existence of \({\fancyscript B}\) -universal functions for admissible sequences. Finally we discuss the problem whether interpolating Blaschke products can ever be \({\fancyscript B}\) -universal for admissible sequences \({(\phi_n)}\).

Boundary interpolation and approximation by infinite Blaschke products

This paper considers the problem of boundary interpolation (in the sense of non-tangential limits) by Blaschke products and interpolating Blaschke products. Simple and constructive proofs, which also work in the more general situation of $H^\infty(\Omega)$ where $\Omega$ is a more general domain, are given of a number of results showing the existence of Blaschke products solving certain interpolation problems at a countable set of points on the circle. A variant of Frostman's theorem is also presented.

Two new characterizations of carleson-newman Blaschke products

We present an extension of a result of Vasyunin by giving a characterization of finite products of interpolating Blaschke products B in terms of the minorization of B(z) by the distance of z to the zeros of B. We also characterize those Blaschke products that satisfy the hereditary weak embedding property.

On Interpolating Blaschke Products and Blaschke-Oscillatory Equations

This research is partially a continuation of a 2007 paper by the author. Growth estimates for generalized logarithmic derivatives of Blaschke products are provided under the assumption that the zero sequences are either uniformly separated or exponential. Such Blaschke products are known as interpolating Blaschke products. The growth estimates are then proven to be sharp in a rather strong sense. The sharpness discussion yields a solution to an open problem posed by E. Fricain and J. Mashreghi in 2008. Finally, several aspects are pointed out to illustrate that interpolating Blaschke products appear naturally in studying the oscillation of solutions of a differential equation f″+A(z)f=0, where A(z) is analytic in the unit disc. In particular, a unit disc analogue of a 1988 result due to S. Bank on prescribed zero sequences for entire solutions is obtained, and a more careful analysis of a 1955 example due to B. Schwarz on the case \(A(z)=\frac{1+4\gamma^{2}}{(1-z^{2})^{2}}\) reveals that an infinite zero sequence is always a union of two exponential sequences.

Blaschke products in Lipschitz spaces

AbstractWe study the membership of Blaschke products in Lipschitz spaces, especially for interpolating Blaschke products and for those whose zeros lie in a Stolz angle. We prove several theorems that complement or extend the earlier works of Ahern and the author.

Cyclic Blaschke products for composition operators

In this work, cyclic Blaschke products for composition operators induced by disc automorphisms are studied. In particular, we obtain interpolating Blaschke products that are cyclic for nonelliptic automorphisms and we obtain a new characterization of Blaschke products that are not finite products of interpolating Blaschke products.

On Blaschke products with derivatives in Bergman spaces with normal weights

We generalize a well-known sufficient condition for interpolating sequences for the Hilbert Bergman spaces to other Bergman spaces with normal weights (as defined by Shields and Williams) and obtain new results regarding the membership of the derivative of a Blaschke product or a general inner function in such spaces. We also apply duality techniques to obtain further results of this type and obtain new results about interpolating Blaschke products.