On démontre un résultat de dichotomie pour les fonctions qui opèrent sur les restrictions d’algèbres de fonctions holomorphes de plusieurs variables. On obtient ce résultat après étude de la séparation par des fonctions holomorphes de compacts sur certaines hypersurfaces de $C^n$.

Let μ and λ be bounded positive singular measures on the unit circle such that μ ⊥ λ. It is proved that there exist positive measures μ₀ and λ₀ such that μ₀ ∼ μ, λ₀ ∼ λ, and $|{\psi }_{\mu ₀}|<1\cap |{\psi }_{\lambda ₀}|<1=\varnothing$, where ${\psi }_{\mu }$ is the associated singular inner function of μ. Let $\left(\mu \right)={\bigcap }_{\nu ;\nu \sim \mu }Z\left({\psi }_{\nu }\right)$ be the common zeros of equivalent singular inner functions of ${\psi }_{\mu }$. Then (μ) ≠ ∅ and (μ) ∩ (λ) = ∅. It follows that μ ≪ λ if and only if (μ) ⊂ (λ). Hence (μ) is the set in the maximal ideal space of $H^{\infty }$ which relates naturally to the set of measures equivalent to μ....

We study connected components of a common zero set of equivalent singular inner functions in the maximal ideal space of the Banach algebra of bounded analytic functions on the open unit disk. To study topological properties of zero sets of inner functions, we give a new type of factorization theorem for inner functions.

Compact composition operators on $H^{\infty }\left(G\right)$, where G is a region in the complex plane, and the spectra of these operators were described by D. Swanton ( Compact composition operators on B(D), Proc. Amer. Math. Soc. 56 (1976), 152-156). In this short note we characterize all compact endomorphisms, not necessarily those induced by composition operators, on $H^{\infty }\left(D\right)$, where D is the unit disc, and determine their spectra.

We prove that every weakly compact multiplicative linear continuous map from $H^{\infty }\left(D\right)$ into $H^{\infty }\left(D\right)$ is compact. We also give an example which shows that this is not generally true for uniform algebras. Finally, we characterize the spectra of compact composition operators acting on the uniform algebra $H^{\infty }\left(B_E\right)$, where $B_E$ is the open unit ball of an infinite-dimensional Banach space E.

We look at normed spaces of differentiable functions on compact plane sets, including the spaces of infinitely differentiable functions considered by Dales and Davie in [7]. For many compact plane sets the classical definitions give rise to incomplete spaces. We introduce an alternative definition of differentiability which allows us to describe the completions of these spaces. We also consider some associated problems of polynomial and rational approximation.

Let Ω be a C∞-domain in Cn. It is well known that a holomorphic function on Ω behaves twice as well in complex tangential directions (see [GS] and [Kr] for instance). It follows from well known results (see [H], [RS]) that some converse is true for any kind of regular functions when Ω satisfies(P)    The real tangent space is generated by the Lie brackets of real and imaginary parts of complex tangent vectorsIn this paper we are interested in the behavior of holomorphic Hardy-Sobolev functions in...

The Hilbert matrix acts on Hardy spaces by multiplication with Taylor coefficients. We find an upper bound for the norm of the induced operator.

We introduce a method to compute rigorous component-wise enclosures of discrete convolutions using the fast Fourier transform, the properties of Banach algebras, and interval arithmetic. The purpose of this new approach is to improve the implementation and the applicability of computer-assisted proofs performed in weighed ${\ell }^1$ Banach algebras of Fourier/Chebyshev sequences, whose norms are known to be numerically unstable. We introduce some application examples, in particular a rigorous aposteriori...

This paper characterizes the Banach algebras of continuous functions on which the spectral factorization mapping 𝔖 is continuous or bounded. It is shown that 𝔖 is continuous if and only if the Riesz projection is bounded on the algebra, and that 𝔖 is bounded only if the algebra is isomorphic to the algebra of continuous functions. Consequently, 𝔖 can never be both continuous and bounded, on any algebra under consideration.