Corona Theorem Research Articles

The invertibility of convolution type operators on a union of intervals and the corona theorem

The invertibility of convolution type operators on unions of intervals is studied. Sufficient conditions of invertibility for some classes of these operators are established. Solvability results forn-term corona problems are obtained using two different approaches: one involving reduction ton−1 Riemann-Hilbert problems in two variables and another involving reduction to two-term corona problems. The invertibility of the convolution operators on a union of intervals is also related to the invertibility of associated convolution operators on single intervals. Formulas for the inverse operators are given.

A New Estimate for the Vector Valued Corona Problem

We improve an estimate of Fuhrmann and Vasyunin in the vector valued corona theorem.

Spherical contractions and interpolation problems on the unit ball

In this note fractional representations of multipliers on vector-valued functional Hilbert spaces are used to give a proof of Arveson's version of von Neumann's inequality for n-contractions on the unit ball. We prove a commutant lifting theorem for operators on the classical Hardy space over the unit ball in \mathbb{C}^{n}. As applications we obtain interpolation results for functions in the Schur class, we deduce a Toeplitz corona theorem on the unit ball, and we give a simplified definition of Arveson's curvature invariant for n-contractions with finite-dimensional defect space. In the final part we describe a solution of the operator-valued Nevanlinna-Pick problem with uniform bounds on uniqueness sets in the unit ball.

Blaschke inductive limits of uniform algebras

We consider and study Blaschke inductive limit algebrasA(b), defined as inductive limits of disc algebras A(D) linked by a sequence of finite Blaschke products. It is well known that big G‐disc algebras AG over compact abelian groups G with ordered duals can be expressed as Blaschke inductive limit algebras. Any Blaschke inductive limit algebra A(b) is a maximal and Dirichlet uniform algebra. Its Shilov boundary ∂A(b) is a compact abelian group with dual group that is a subgroup of ℚ. It is shown that a big G‐disc algebra AG over a group G with ordered dual is a Blaschke inductive limit algebra if and only if . The local structure of the maximal ideal space and the set of one‐point Gleason parts of a Blaschke inductive limit algebra differ drastically from the ones of a big G‐disc algebra. These differences are utilized to construct examples of Blaschke inductive limit algebras that are not big G‐disc algebras. A necessary and sufficient condition for a Blaschke inductive limit algebra to be isometrically isomorphic to a big G‐disc algebra is found. We consider also inductive limits H∞(I) of algebras H∞, linked by a sequence of inner functions, and prove a version of the corona theorem with estimates for it. The algebra H∞(I) generalizes the algebra of bounded hyper‐analytic functions on an open big G‐disc, introduced previously by Tonev.

Behavioral controllability of time-delay systems with incommensurable delays

In this paper it is shown that for time-delay systems in pseudo-state form with (in)commensurable time-delays, the notion of behavioral controllability and spectral controllability are equivalent. This yields a generalized Haut us test for behavioral controllability of time-delay systems. The result is obtained by considering delay systems as convolution systems, and relies on a result of Hörmander, that generalizes the Corona Theorem to the Paley-Wiener algebra of holomorphic functions of exponential type that are polynomially bounded on the real axis.

Matrix-valued corona theorem for multiply connected domains

Let D ⊂ C be a bounded domain, whose boundary B consists of k simple closed continuous curves and H ∞ (D) be the algebra of bounded analytic functions on D. We prove the matrix-valued corona theorem for matrices with entries in H ∞ (D).

On the corona theorem for almost periodic functions

LetAPΣ+(Rn) denote the Banach algebra of all continuous almost periodic functions onRn whose Bohr-Fourier spectrum is contained in an additive semi-group Σ⊂[0, ∞)n. We show that the maximal ideal space ofAPΣ+(Rn) may have a nonempty corona and we characterize all Σ for which the corona is empty. Analogous results are established for algebras of almost periodic functions with absolutely convergent Fourier series.

Sarason interpolation and Toeplitz corona theorem for almost periodic matrix functions

Sarason interpolation and Toeplitz corona problems are studied for almost periodic matrix functions. Recent results on almost periodic factorization and related generalized Toeplitz operators are the main tools in the study.

A new construction of Riemann surfaces with corona

f lg l + . . . + fngn =1 ; equivalently [8, VIII.2], the corona .AA(X) \ t(X) is empty. (Here .All(X) is the maximal ideal space of the algebra H ~ (X) of bounded holomorphic functions on X and t is the natural inclusion X '-+ .M(X).) If X does not satisfy the corona theorem, then X may be said to have corona. Riemann surfaces known to satisfy the corona theorem include the unit disk [3], bordered Riemann surfaces [1, 16], and various classes of planar domains [10, 14]. The question of whether general planar domains satisfy the corona theorem is open. The first construction of a Riemann surface with corona is due to Cole [7]. The goal of this paper is to prove the following.

Unitary Colligations, Reproducing Kernel Hilbert Spaces, and Nevanlinna–Pick Interpolation in Several Variables

Recently J. Agler studied the class Sdof scalar-valued, analytic functions ofdcomplex variablesffor whichf(T1, …, Td) has norm at most 1 for any collection ofdcommuting contractions (T1, …, Td) on a Hilbert space H. Among other results he obtained a characterization of such functions in terms of a positivity property and in terms of a representation as the transfer function of a certain type of d-variable linear system, as well as a Nevanlinna–Pick interpolation theorem for this class of functions. In this note we examine the system theory aspects and uniqueness of the transfer function representation, and give a simpler proof of the Nevanlinna–Pick interpolation theorem for the class Sdand obtain ad-variable version of the Toeplitz corona theorem. By using ideas of Arov and Grossman introduced for 1-variable problems, as a bonus we obtain a collection of linear fractional maps which parametrize the set of all Sdsolutions of an interpolation problem.

The Corona Theorem and the Existence of Canonical Factorization of Triangular AP-Matrix Functions

Using the corona theorem, canonical factorizability criteria for triangular almost periodic matrix-valued functions of the form[formula]are obtained. This allows the authors to obtain effective criteria for the existence of canonical factorization as well as to construct explicit factorizations for new classes of almost periodic matrix functions in a companion paper.

The Corona Theorem and the Canonical Factorization of Triangular AP Matrix Functions—Effective Criteria and Explicit Formulas

Using results of the companion paper, the authors obtain effective criteria for the existence of canonical factorization for new classes of triangular almost periodic matrix functions of the form[formula]. The important role in the proof of necessity in these criteria is played by a result on the stability of canonical AP factorizability obtained in the present paper. By finding corona solutions for a pair of binomial almost periodic functions, it is possible to calculate explicitly the factors of a canonical factorization for a class of considered matrix functions with trinomialf.

Almost periodic factorization and corona theorem

Almost periodic factorization and corona theorem

Bounded Functions in Möbius Invariant Dirichlet Spaces

Forp∈(0,1), letQp(Qp,0) be the space of analytic functionsfon the unit diskΔwith supw∈Δ‖f∘ϕw‖Dp<∞ (lim|w|→1‖f∘ϕw‖Dp=0), where ‖·‖Dpmeans the weighted Dirichlet norm andϕwis the Möbius map ofΔonto itself withϕw(0)=w. In this paper, we prove the Corona theorem for the algebraQp∩H∞(Qp,0∩H∞); then we provide a Fefferman–Stein type decomposition forQp(Qp,0), and finally we describe the interpolating sequences forQp∩H∞(Qp,0∩H∞)).

The Hp corona theorem in analytic polyhedra

TheHp corona problem is the following: Letg1, ...,gm be bounded holomorphic functions with 0<δ≤Σ‖gi‖. Can we, for anyHp function ϕ, findHp functionsu1, ...,um such that Σgiui=ϕ? It is known that the answer is affirmative in the polydisc, and the aim of this paper is to prove that it is in non-degenerate analytic polyhedra. To prove this, we construct a solution using a certain integral representation formula. TheHp estimate for the solution is then obtained by localization and some harmonic analysis results in the polydisc.

A division problem for $$\bar \partial _b - CLOSED$$ forms

LetG 1,…,Gm be bounded holomorphic functions in a strictly pseudoconvex domainD such that\(\delta ^2 \leqslant \sum {\left| {G_j } \right|^2 \leqslant 1} \). We prove that for each\(\bar \partial _b - closed\) (0,q)-form ϕ inL p(∂D), 1<p<∞, there are\(\bar \partial _b - closed\) formsu 1, …,u m inL p(∂D) such that ΣG juj=ϕ. This generalizes previous results forq=0. The proof consists in delicate estimates of integral representation formulas of solutions and relies on a certainT1 theorem due to Christ and Journe. For (0,n−1)-forms there is a simpler proof that also gives the result forp=∞. Restricted to one variable this is precisely the corona theorem.

The generalized corona theorem

The generalized corona theorem

Flows in Fibers

Let ${H^\infty }(\Delta )$ be the algebra of all bounded analytic functions on the open unit disc $\Delta$, and let $\mathfrak {M}({H^\infty }(\Delta ))$ be the maximal ideal space of ${H^\infty }(\Delta )$. Using a flow, we represent a reasonable portion of a fiber in $\mathfrak {M}({H^\infty }(\Delta ))$. This indicates a relation between the corona theorem and the individual ergodic theorem. As an application, we answer a question of Forelli [3] by showing that there exists a minimal flow on which the induced uniform algebra is not a Dirichlet algebra. The proof rests on the fact that the closure of a nonhomeomorphic part in $\mathfrak {M}({H^\infty }(\Delta ))$ may contain a homeomorphic copy of $\mathfrak {M}({H^\infty }(\Delta ))$. Taking suitable factors, we may construct a lot of minimal flows which are not strictly ergodic.

Flows in fibers

Let H ∞ ( Δ ) {H^\infty }(\Delta ) be the algebra of all bounded analytic functions on the open unit disc Δ \Delta , and let M ( H ∞ ( Δ ) ) \mathfrak {M}({H^\infty }(\Delta )) be the maximal ideal space of H ∞ ( Δ ) {H^\infty }(\Delta ) . Using a flow, we represent a reasonable portion of a fiber in M ( H ∞ ( Δ ) ) \mathfrak {M}({H^\infty }(\Delta )) . This indicates a relation between the corona theorem and the individual ergodic theorem. As an application, we answer a question of Forelli [3] by showing that there exists a minimal flow on which the induced uniform algebra is not a Dirichlet algebra. The proof rests on the fact that the closure of a nonhomeomorphic part in M ( H ∞ ( Δ ) ) \mathfrak {M}({H^\infty }(\Delta )) may contain a homeomorphic copy of M ( H ∞ ( Δ ) ) \mathfrak {M}({H^\infty }(\Delta )) . Taking suitable factors, we may construct a lot of minimal flows which are not strictly ergodic.

The 𝐻² corona problem and \overline∂_{𝑏} in weakly pseudoconvex domains

We derive a Bochner-Kodaira-Nakano-Morrey-Kohn-Hörmander type equality in holomorphic vector bundles and obtain L 2 {L^2} -estimates for ∂ ¯ b {\bar \partial _b} in a pseudoconvex domain that admits a plurisubharmonic C 2 {C^2} defining function. We combine these with the trick in Wolff’s proof of the corona theorem and obtain a H 2 {H^2} -corona theorem in such a domain.