Let $f$ be a continuous map of the closure $\overline{\mathrm{\Delta }}$ of the open unit disc $\mathrm{\Delta }$ of $\mathbb{C}$ into a unital associative Banach algebra $\mathcal{A}$, whose restriction to $\mathrm{\Delta }$ is holomorphic, and which satisfies the condition whereby $0\notin \sigma \left(f\left(z\right)\right)\subset \overline{\mathrm{\Delta }}$ for all $z\in \mathrm{\Delta }$ and $\sigma \left(f\left(z\right)\right)\subset \partial \mathrm{\Delta }$ whenever $z\in \partial \mathrm{\Delta }$ (where $\sigma \left(x\right)$ is the spectrum of any $x\in \mathcal{A}$). One of the basic results of the present paper is that $f$ is , that is to say, $\sigma \left(f\left(z\right)\right)$ is then a compact subset of $\partial \mathrm{\Delta }$ that does not depend on $z$ for all $z\in \overline{\mathrm{\Delta }}$. This fact will be applied to holomorphic self-maps of the open unit ball of some $J^{*}$-algebra...

Let $D^{\text{'}}\subset {ℂ}^{n-1}$ be a bounded domain of Lyapunov and $f(z^{\text{'}},z_n)$ a holomorphic function in the cylinder $D=D^{\text{'}}\times U_n$ and continuous on $\overline{D}$. If for each fixed $a^{\text{'}}$ in some set $E\subset \partial D^{\text{'}}$, with positive Lebesgue measure $\text{mes}E>0$, the function $f(a^{\text{'}},z_n)$ of $z_n$ can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then $f(z^{\text{'}},z_n)$ can be holomorphically continued to $(D^{\text{'}}\times ℂ)\setminus S$, where $S$ is some analytic (closed pluripolar) subset of $D^{\text{'}}\times ℂ$.

The space of multilinear mappings of nuclear type (s;r1,...,rn) between Banach spaces is considered, some of its properties are described (including the relationship with tensor products) and its topological dual is characterized as a Banach space of absolutely summing mappings.

Si studiano «combinazioni convesse complesse» per mappe olomorfe dal disco unità di $ℂ$ in un dominio convesso limitato $D$ di uno spazio di Banach complesso $E$, e se ne traggono conseguenze sul carattere globale della non unicità per le geodetiche complesse di $D$.

It is shown that an infinite sequence of polynomial mappings of several complex variables, with suitable growth restrictions, determines a filled-in Julia set which is pluriregular. Such sets depend continuously and analytically on the generating sequences, in the sense of pluripotential theory and the theory of set-valued analytic functions, respectively.

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...

We consider scalar products on a given Hilbert space parametrized by bounded positive and invertible operators defined on this space, and orthogonal projectors onto a fixed closed subspace of the initial Hilbert space corresponding to these scalar products. We show that the projector is an analytic function of the scalar product, we give the explicit formula for its Taylor expansion, and we prove some algebraic formulas for projectors.