Disk Algebra Research Articles

Exponential factorizations of holomorphic maps

We show that any element of the special linear group $SL_2(R)$ is a product of two exponentials if the ring $R$ is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential factor is not enough since the exponential map corresponding to $SL_2(\mathbb{C})$ is not surjective. Our result extends to the linear group $GL_2(R)$.

Strong algebrability and residuality on certain sets of analytic functions

We show that the set of analytic functions from $\mathbb C^2$ into $\mathbb C^2$, which are not Lorch-analytic is spaceable and strongly $\mathfrak {c}$-algebrable, but is not residual in the space of entire functions from $\mathbb C^2$ into $\mathbb C^2$. We also show that the set of functions which belongs to the disk algebra but not a Dales-Davie algebra is strongly $\mathfrak {c}$-algebrable and is residual in the disk algebra.

On universality and convergence of the Fourier series of functions in the disc algebra

We construct functions in the disc algebra whose Fourier series are pointwise universal on countable and dense sets and their sets of divergence contain Gδ and dense sets and have Hausdorff dimension zero. We also see that some classes of closed sets of measure zero do not accept uniformly universal Fourier series, although all such sets accept divergent Fourier series.

Taylor sums for functions in the disk algebra

Taylor sums for functions in the disk algebra

Regularity for generators of invariant subspaces of the Dirichlet shift

Let D denote the classical Dirichlet space of analytic functions on the open unit disc whose derivative is square area integrable. For a set E⊆∂D we writeDE={f∈D:limr→1−⁡f(reit)=0q.e.}, where q.e. stands for “except possibly for eit in a set of logarithmic capacity 0”. We show that if E is a Carleson set, then there is a function f∈DE that is also in the disc algebra and that generates DE in the sense that DE=clos{pf:p is a polynomial}.We also show that if φ∈D is an extremal function (i.e. 〈pφ,φ〉=p(0) for every polynomial p), then the limits of |φ(z)| exist for every eit∈∂D as z approaches eit from within any polynomially tangential approach region.

Completely Contractive Extensions of Hilbert Modules Over Tensor Algebras

This paper studies completely contractive extensions of Hilbert modules over tensor algebras over $$C^*$$ -correspondences. Using a result of Sz-Nagy and Foias on triangular contractions, extensions are parametrized in terms of contractive intertwining maps between certain defect spaces. These maps have a simple description when initial data consists of partial isometries. Sufficient conditions for the vanishing and nonvanishing of completely contractive Hilbert module $${\text {Ext}}^1$$ are given that parallel results for the classical disc algebra.

Carlson's theorem for different measures

We use an observation of Bohr connecting Dirichlet series in the right half plane C+ to power series on the polydisk to interpret Carlson's theorem about integrals in the mean as a special case of the ergodic theorem by considering any vertical line in the half plane as an ergodic flow on the polytorus. Of particular interest is the imaginary axis because Carlson's theorem for Lebesgue measure does not hold there. In this note, we construct measures for which Carlson's theorem does hold on the imaginary axis for functions in the Dirichlet series analog of the disk algebra A(C+).

Rational dilation problems associated with constrained algebras

A set Ω is a spectral set for an operator T if the spectrum of T is contained in Ω, and von Neumann's inequality holds for T with respect to the algebra R(Ω) of rational functions with poles off of Ω‾. It is a complete spectral set if for all r∈N, the same is true for Mr(C)⊗R(Ω). The rational dilation problem asks, if Ω is a spectral set for T, is it a complete spectral set for T? There are natural multivariable versions of this. There are a few cases where rational dilation is known to hold (eg, over the disk and bidisk), and some where it is known to fail, for example over the Neil parabola, a distinguished variety in the bidisk. The Neil parabola is naturally associated to a constrained subalgebra of the disk algebra C+z2A(D). Here it is shown that such a result is generic for a large class of varieties associated to constrained algebras. This is accomplished in part by finding a minimal set of test functions. In addition, an Agler–Pick interpolation theorem is given and it is proved that there exist Kaijser–Varopoulos style examples of non-contractive unital representations where the generators are contractions.

Aleksandrov–Clark Theory for Drury–Arveson Space

Recent work has demonstrated that Clark’s theory of unitary perturbations of the backward shift on a deBranges–Rovnyak space on the disk has a natural extension to the several-variable setting. In the several-variable case, the appropriate generalization of the Schur class of contractive analytic functions is the closed unit ball of the Drury–Arveson multiplier algebra and the Aleksandrov–Clark measures are necessarily promoted to positive linear functionals on a symmetrized subsystem of the Free Disk operator system $$\mathcal {A} _d + \mathcal {A} _d ^*$$ , where $$\mathcal {A} _d$$ is the Free or Non-commutative Disk Algebra on d generators. We continue this program for vector-valued Drury–Arveson space by establishing the existence of a canonical ‘tight’ extension of any Aleksandrov–Clark map to the full Free Disk operator system. We apply this tight extension to generalize several earlier results and we characterize all extensions of the Aleksandrov–Clark maps.

On the boundary behaviour of derivatives of functions in the disc algebra

We provide with a simple and explicit construction of a function in the disc algebra, whose derivatives enjoy a disjoint universal property near the boundary. The set of functions with such property is topologically generic, densely lineable, and spaceable.

Boundary-Nonregular Functions in the Disc Algebra and in Holomorphic Lipschitz Spaces

We prove in this paper the existence of dense linear subspaces in the classical holomorphic Lipschitz spaces in the disc all of whose non-null functions are nowhere differentiable at the boundary. Infinitely generated free algebras as well as infinite dimensional Banach spaces consisting of Lipschitz functions enjoying the mentioned property almost everywhere on the boundary are also exhibited. It is also investigated the algebraic size of the family of functions in the disc algebra that either do not preserve Borel sets on the unit circle or possess the Cantor boundary behavior on the disc.

On Weierstrass' monsters in the disc algebra

Let $\Omega$ be a Jordan domain in the complex plane whose boundary is piecewise analytic, and let $A(\Omega )$ be the algebra of all holomorphic functions on $\Omega$ that are continuous up to the boundary. We prove the existence of dense linear subspaces and of infinitely generated subalgebras in $A(\Omega )$ all of whose nonzero members are, in a strong sense, not differentiable at almost any point of the boundary. We also obtain infinite-dimensional closed subspaces consisting of functions that are not differentiable at any point of a dense subset of the boundary. In the case of the unit disc, those dense linear subspaces can be found with their functions being nowhere differentiable in the unit circle.

Nowhere hölderian functions and Pringsheim singular functions in the disc algebra

We prove the existence of dense linear subspaces, of infinitely generated subalgebras and of infinite dimensional Banach spaces in the disc algebra all of whose nonzero members are not $$\alpha $$ -holderian at any point of the unit circle for any $$\alpha >0$$ . This completes the recently established result of topological genericity of this kind of functions, as well as the corresponding lineability statements about functions that are nowhere differentiable at the boundary. Topological and algebraic genericity is also studied for the family of boundary-smooth holomorphic functions that are Pringsheim singular at any point of the unit circle.

Generic Behavior of Classes of Taylor Series Outside the Unit Disk

It is known that, generically, Taylor series of functions holomorphic in the unit disk turn out to be “maximally divergent” outside of the disk. For functions in classical Banach spaces of holomorphic functions, as for example, Hardy spaces or the disk algebra, the situation is more delicate. In this paper, it is shown that the behavior of the partial sums on sets outside the open unit disk sensitively depends on the way the sets touch the unit circle. As main tools, results on simultaneous approximation by polynomials are proved.

Ideals in a multiplier algebra on the ball

We study the ideals of the closure of the polynomial multipliers on the Drury-Arveson space. Structural results are obtained by investigating the relation between an ideal and its weak- ∗ * closure, much in the spirit of the corresponding classical facts for the disc algebra. Zero sets for multipliers are also considered and are deeply intertwined with the structure of ideals. Our approach is primarily based on duality arguments.

Multiplier algebras of complete Nevanlinna–Pick spaces: Dilations, boundary representations and hyperrigidity

We study reproducing kernel Hilbert spaces on the unit ball with the complete Nevanlinna–Pick property through the representation theory of their algebras of multipliers. We give a complete description of the representations in terms of the reproducing kernels. While representations always dilate to ⁎-representations of the ambient C⁎-algebra, we show that in our setting we automatically obtain coextensions. In fact, we show that in many cases, this phenomenon characterizes the complete Nevanlinna–Pick property. We also deduce operator theoretic dilation results which are in the spirit of work of Agler and several other authors. Moreover, we identify all boundary representations, compute the C⁎-envelopes and determine hyperrigidity for certain analogues of the disc algebra. Finally, we extend these results to spaces of functions on homogeneous subvarieties of the ball.

The commutant of a multiplication operator with a finite Blaschke product symbol on the Sobolev disk algebra

Let R(D) be the algebra generated in the Sobolev space W22(D) by the rational functions with poles outside the unit disk D¯. This is called the Sobolev disk algebra. In this article, the commutant of the multiplication operator MB(z) on R(D) is studied, where B(z) is an n-Blaschke product. We prove that an operator A∈L(R(D)) is in A'(MB(z)) if and only if A=∑i=1nMhiMΔ(z)−1Ti, where {hi}i=1n⊂R(D), and Ti∈L(R(D)) is given by (Tig)(z)=∑j=1n(−1)i+jΔij(z)g(Gj−1(z)), i=1,2,…,n, G0(z)≡z.

Non-commutative Clark measures for the free and Abelian Toeplitz algebras

We construct a non-commutative Aleksandrov–Clark measure for any element in the operator-valued free Schur class, the closed unit ball of the free Toeplitz algebra of vector-valued full Fock space over Cd. Here, the free (analytic) Toeplitz algebra is the unital weak operator topology (WOT)-closed algebra generated by the component operators of the free shift, the row isometry of left creation operators. This defines a bijection between the free operator-valued Schur class and completely positive maps (non-commutative AC measures) on the operator system of the free disk algebra, the norm-closed algebra generated by the free shift. Identifying Drury–Arveson space with symmetric Fock space, we determine the relationship between the non-commutative AC measures for elements of the operator-valued commutative Schur class (the closed unit ball of the WOT-closed Toeplitz algebra generated by the Arveson shift) and the AC measures of their free liftings to the free Schur class.

Logarithms and Exponentials in the Matrix Algebra $${\mathcal M}_2(A)$$ M 2 ( A )

It is well known that in the disk algebra $$A({ \mathbb D})$$ every zero-free function has a logarithm in $$A({ \mathbb D})$$ . This is no longer true if we look at invertible matrices over $$A({ \mathbb D})$$ . In this paper, we give a sufficient condition on the trace of a $$2\times 2$$ -matrix M so that $$M=e^L$$ for some matrix $$L\in A({ \mathbb D})$$ . We compute all the logarithms of the identity matrix in $${\mathcal M}_2(A({ \mathbb D}))$$ and observe that the anti-diagonal elements can be arbitrarily prescribed. We also characterize those upper (or lower) triangular matrices which are exponentials in $${\mathcal M}_2(A({ \mathbb D}))$$ and determine all their logarithms. This will enable us to prove that $$\exp {\mathcal M}_2(A({ \mathbb D}))$$ is neither closed nor open within the principal component of $${\mathcal M}_2(A({ \mathbb D}))^{-1}$$ . Finally, we show that every invertible matrix in $${\mathcal M}_2(A({ \mathbb D}))$$ is a product of four exponential matrices and give conditions for reducing this number. These results will be put into the more general setting of commutative Banach algebras whenever possible.

Reducibility and unitary equivalence of analytic multipliers on Sobolev disk algebra

It is proved that Mϕ with 2-Blaschke product ϕ is reducible on Sobolev disk algebra R(D) if and only if ϕ=βα−z21−α‾z2 for some α∈D and unimodular constant β. Also, an analytic multiplier Mϕ is unitary equivalent to Mzn on R(D) if and only if ϕ=λzn for some unimodular constant λ.