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We prove the reducibility of analytic multipliers Mϕ with a class of finite Blaschke products symbol ϕ on the Sobolev disk algebra R(ⅅ). We also describe their nontrivial minimal reducing subspaces.

This article considers the problem of optimally recovering stable linear time-invariant systems observed via linear measurements made on their transfer functions. A common modeling assumption is replaced here by the related assumption that the transfer functions belong to a model set described by approximation capabilities. Capitalizing on recent optimal-recovery results relative to such approximability models, we construct some optimal algorithms and characterize the optimal performance for the identification and evaluation of transfer functions in the framework of the Hardy Hilbert space and of the disc algebra. In particular, we determine explicitly the optimal recovery performance for frequency measurements taken at equispaced points on an inner circle or on the torus., Comment: 22 pages, 2 figures

We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $$(C,\alpha )$$ -summable to the function in the norm of the space, provided that $$\alpha >1/2$$ . We further show that the constant 1/2 is sharp, in marked contrast with the classical case of the disk algebra.

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.

It is proved that M ϕ with 2-Blaschke product ϕ is reducible on Sobolev disk algebra R ( D ) if and only if ϕ = β α − z 2 1 − α ‾ z 2 for some α ∈ D and unimodular constant β . Also, an analytic multiplier M ϕ is unitary equivalent to M z n on R ( D ) if and only if ϕ = λ z n for some unimodular constant λ .

We prove that functions analytic in the unit disk and continuous up to the boundary are dense in the de Branges–Rovnyak spaces induced by the extreme points of the unit ball of H ∞ . Together with previous theorems, it follows that this class of functions is dense in any de Branges–Rovnyak space.

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.

We study [ φ t , X ] , the maximal space of strong continuity for a semigroup of composition operators induced by a semigroup { φ t } t ≥ 0 of analytic self-maps of the unit disk, when X is BMOA, H ∞ or the disk algebra. In particular, we show that [ φ t , BMOA ] ≠ BMOA for all nontrivial semigroups. We also prove, for every semigroup { φ t } t ≥ 0 , that lim t → 0 + ⁡ φ t ( z ) = z not just pointwise, but in H ∞ norm. This provides a unified proof of known results about [ φ t , X ] when X ∈ { H p , A p , B 0 , VMOA } .

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.

Let T T be an operator on a Hilbert space H H with numerical radius w ( T ) ≤ 1 w(T)\le 1 . According to a theorem of Berger and Stampfli, if f f is a function in the disk algebra such that f ( 0 ) = 0 f(0)=0 , then w ( f ( T ) ) ≤ ‖ f ‖ ∞ w(f(T))\le \|f\|_\infty . We give a new and elementary proof of this result using finite Blaschke products. A well-known result relating numerical radius and norm says ‖ T ‖ ≤ 2 w ( T ) \|T\| \leq 2w(T) . We obtain a local improvement of this estimate, namely, if w ( T ) ≤ 1 w(T)\le 1 , then \[ ‖ T x ‖ 2 ≤ 2 + 2 1 − | ⟨ T x , x ⟩ | 2 ( x ∈ H , ‖ x ‖ ≤ 1 ) . \|Tx\|^2\le 2+2\sqrt {1-|\langle Tx,x\rangle |^2} \qquad (x\in H,~\|x\|\le 1). \] Using this refinement, we give a simplified proof of Drury’s teardrop theorem, which extends the Berger–Stampfli theorem to the case f ( 0 ) ≠ 0 f(0)\ne 0 .

Given a closed set E of Lebesgue measure zero on the unit circle T there is a continuous function f on T such that for every continuous function g on E there is a subsequence of partial Fourier sums S n + ( f , ζ ) = ∑ k = 0 n f ˆ ( k ) ζ k of f , which converges to g uniformly on E . This result completes one result in a recent paper by C. Papachristodoulos and M. Papadimitrakis (2019), see Papachristodoulos and Papadimitrakis (2019). They proved that for a classical one third Cantor set C there is no universal function in the disk algebra. They also proved that for a symmetric Cantor set C ∗ on T there is no universal continuous function for the classical symmetric Fourier sums. See also [2] .

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.

It is shown that rational dilation fails on broad collection of distinguished varieties associated to constrained subalgebras of the disk algebra of the form C + B A(D), where B is a finite Blaschke product with two or more zeros. 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., Page proof corrections included in this version!

Let $R(\mathbb{D})$ be the algebra generated in the Sobolev space $W^{22}(\mathbb{D})$ by the rational functions with poles outside the unit disk $\overline{\mathbb{D}}$ . This is called the Sobolev disk algebra. In this article, the commutant of the multiplication operator $M_{B(z)}$ on $R(\mathbb{D})$ is studied, where $B(z)$ is an n-Blaschke product. We prove that an operator $A\in\mathcal{L}(R(\mathbb{D}))$ is in $\mathcal{A}'(M_{B(z)})$ if and only if $A=\sum_{i=1}^{n}M_{h_{i}}M_{\Delta(z)}^{-1}T_{i}$ , where $\{h_{i}\}_{i=1}^{n}\subset R(\mathbb{D})$ , and $T_{i}\in\mathcal{L}(R(\mathbb{D}))$ is given by $(T_{i}g)(z)=\sum_{j=1}^{n}(-1)^{i+j}\Delta_{ij}(z)g(G_{j-1}(z))$ , $i=1,2,\ldots,n$ , $G_{0}(z)\equiv z$ .

This paper considers the classical problem of recovering functions in the disk algebra from discrete samples taken at the unit circle by means of linear sampling type approximation methods. The paper characterizes subspaces of the disk algebra on which linear approximation operators do exist and subspaces on which such operators do not exist. The subspaces on which a linear signal recovery is possible are characterized by a sufficient concentration of the signal energy in the low frequency components of the signal.

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 + ) .

[EN] In this paper we give a new characterization of when a Banach space E has the Schur property in terms of the disk algebra. We prove that E has the Schur property if and only if A(D, E) = A(D,E-w). (C) 2016 Elsevier Inc. All rights reserved., The first and third authors were supported by MINECO MTM2014-57838-C2-2-P and Prometeo II/2013/013. The second author was supported by MINECO MTM2013-43540-P.

We will characterize the boundedness and compactness of weighted composition operators on the closed subalgebra H∞ ∩ $\mathcal{B}$o between the disk algebra and the space of bounded analytic functions on the open unit disk.

Let be the family of analytic functions on . Let be an analytic self-map of and for . Let be the weighted composition operator on . It is proved that if contains the disk algebra, then there is with such that is an automorphism of and on .

Let D be the unit disk and SA ( D ) denote the Sobolev disk algebra which consists of all analytic functions in the Sobolev space W 2 , 2 ( D ) . In this note, we prove that M z n is similar to ⨁ 1 n M z on SA ( D ) . Then using the matrix manipulations combined with operator theory methods, we characterize the reducing subspaces of M z n on SA ( D ) .

It is shown how to embed the polydisk algebras (finite and infinite ones) into the disk algebra $A(\overline{\mathbb D})$. As a consequence, one obtains uniform closed subalgebras of $A(\overline{\mathbb D})$ which have arbitrarily prescribed stable ranks, Comment: 5 pages

In this paper we study some geometrical properties of certain classes of uniform algebras, in particular the ball algebra A u ( B X ) of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space X . We prove that A u ( B X ) has k -numerical index 1 for every k , the lushness and also the AHSP. Moreover, the disk algebra A ( D ) , and more in general any uniform algebra whose Choquet boundary has no isolated points, is proved to have the polynomial Daugavet property. Most of those properties are extended to the vector valued version A X of a uniform algebra A .

Using the facts that the disk algebra and the Wiener algebra are not coherent, we prove that the polydisc algebra, the ball algebra and the Wiener algebra of the polydisc are not coherent., Comment: 3 pages

In this paper, we prove that for every compact set of the unit disk of logarithmic capacity 0, there exists a Schur function both in the disk algebra and in the Dirichlet space such that the associated composition operator is in all Schatten classes (of the Dirichlet space), and for which the set of points whose image touches the unit circle is equal to this compact set. We show that for every bounded composition operator on the Dirichlet space and for every point of the unit circle, the logarithmic capacity of the set of point having this point as image is 0. We show that every compact composition operator on the Dirichlet space is compact on the gaussian Hardy-Orlicz space; in particular, it is in every Schatten class on the usual Hilbertian Hardy space. On the other hand, there exists a Schur function such that the associated composition operator is compact on the gaussian Hardy-Orlicz space, but which is not even bounded on the Dirichlet space. We prove that the Schatten classes on the Dirichlet space can be separated by composition operators. Also, there exists a Schur function such that the associated composition operator is compact on the Dirichlet space, but in no Schatten class.

In this paper, we give a similarity classification for the multiplication operator M g on the Sobolev disk algebra \(R(\mathbb{D})\) with g analytic on the closure of the unit disk \(\mathbb{D}\).

Let COP =0∩H∞, where0is the little Bloch space on the open unit disk, andA() be the disk algebra on. For non-zero functionsu1,u2,. . .,uN∈A() and distinct analytic self-maps ϕ1,ϕ2,. . .,ϕNsatisfying ϕj∈A() and ∥ϕj∥∞=1 for everyj, it is given characterisations of which the sum of weighted composition operators ∑Nj=1ujCϕjmaps COP intoA().