En reprenant le travail de Weinstock concernant l’union de deux sous-espaces, nous montrons que ${ℂ}_n$ peut être obtenu comme l’union d’un nombre fini de sous-espaces vectoriels totalement réels maximaux, pour tout $n$ supérieur à un. Ceci contraste avec le cas des droites complexes de ${ℂ}^2$, dont il faut un ensemble de capacité positive pour que l’enveloppe soit tout l’espace. On étudie aussi le cas des trois plans réels de ${ℂ}^2$ : si les trois unions deux à deux ne sont pas polynomialement convexes, alors l’enveloppe...

We construct a non-polynomially convex compact subset of the unit torus in ${ℂ}^2$ with polynomially convex hull containing no analytic structure.

In this note by using techniques similar to that of [2] and [3], we study the local polynomial convexity of perturbation of union of two totally real planes meeting along a real line.

We investigate some aspects of maximum modulus sets in the boundary of a strictly pseudoconvex domain $D$ of dimension $N$. If $\Sigma \subset bD$ is a smooth manifold of dimension $N$ and a maximum modulus set, then it admits a unique foliation by compact interpolation manifolds. There is a semiglobal converse in the real analytic case. Two functions in $A^2\left(D\right)$ with the same smooth $N$-dimensional maximum modulus set are analytically related and are polynomially related if a certain homology class in $H_1(D,\mathbf{R})$ vanishes or if $\bar{D}\subset {\mathbf{C}}^N$ is polynomially...

Soit $K$ un compact polynomialement convexe de ${\mathbf{C}}^n$ et $V_K$ son “potentiel logarithmique extrémal” dans ${\mathbf{C}}^n$. Supposons que $K$ est régulier (i.e. $V_K$ continue) et soit $f$ une fonction holomorphe sur un voisinage de $K$. On construit alors une suite $\{{\mathbf{P}}_{\ell }{\}}_{\ell \ge 1}$ de polynôme de $n$ variables complexes avec deg$({\mathbf{P}}_{)}\le \ell$ pour $\ell \ge 1$, telle que l’erreur d’approximation ${\max }_{z\in K}|f\left(z\right)-P_{\ell }\left(z\right)|$ soit contrôlée de façon assez précise en fonction du “pseudorayon de convergence” de $f$ par rapport à $K$ et du degré de convergence $\ell$. Ce résultat est ensuite utilisé pour étendre...

For large classes of complex Banach spaces (mainly operator spaces) we consider orbits of finite rank elements under the group of linear isometries. These are (in general) real-analytic submanifolds of infinite dimension but of finite CR-codimension. We compute the polynomial convex hull of such orbits $M$ explicitly and show as main result that every continuous CR-function on $M$ has a unique extension to the polynomial convex hull which is holomorphic in a certain sense. This generalizes to infinite...

Let T be a representation of a suitable abelian semigroup S by isometries on a Banach space. We study the spectral conditions which will imply that T(s) is invertible for each s in S. On the way we analyse the relationship between the spectrum of T, Sp(T,S), and its unitary spectrum $S{p}_u(T,S)$. For $S={\mathbb{Z}}_{+}^n$ or ${ℝ}_{+}^n$, we establish connections with polynomial convexity.

The paper is devoted to the study of polynomially convex hulls of compact subsets of ℂ², fibered over the boundary of the unit disc, such that all fibers are simple arcs in the plane and their endpoints form boundaries of two closed, not intersecting analytic discs. The principal question concerned is under what additional condition such a hull is a bordered topological hypersurface and, in particular, is foliated by a unique holomorphic motion. One of the main results asserts that this happens...

We prove a theorem on the boundary regularity of a purely p-dimensional complex subvariety of a relatively compact, strictly pseudoconvex domain in a Stein manifold. Some applications describing the structure of the polynomial hull of closed curves in Cn are also given.

We use our disc formula for the Siciak-Zahariuta extremal function to characterize the polynomial hull of a connected compact subset of complex affine space in terms of analytic discs.