arXiv subjects
Publications and source records attributed to Sofia Ortega Castillo.
We reduce the polynomial cluster value problem for the algebra of bounded analytic functions, $H^{\infty}$, on the ball of Banach spaces $X$ to the same polynomial cluster value problem for $H^{\infty}$ but on the ball of those spaces which are $\ell_1$-sums of finite dimensional spaces.
Having been unclear how to define that a domain is strictly pseudoconvex in the infinite-dimensional setting, we develop a general theory having Banach spaces in mind. We first focus on finite dimension and eliminate the need of two degrees of differentiability of the boundary of a domain, since differentiable functions are difficult to find in infinite dimension. We introduce $\ell$-strict pseudoconvexity for $\ell\geq 1$, $1$-strict pseudoconvexity at the boundary, $\ell$-uniform pseudoconvexity for $\ell\geq 0$ and finally strong pseudoconvexity. Defining $\ell$-strict pseudoconvexity and $\ell$-uniform pseudoconvexity for $\ell<2$ depends on extending a notion of strict plurisubharmonicity to cases lacking $C^2$-smoothness, first studying it in the sense of distribution and then considering it in infinite dimension. Examples of strictly plurisubharmonic functions as well as strongly pseudoconvex domains are presented, which end up related to important classical Banach spaces. Finally, some solutions to the inhomogeneous Cauchy-Riemann equations for $\overline{\partial}$-closed $(0,1)$-forms in infinite-dimensional domains are shown, giving new information about domains affinely isomorphic to the ball of $\ell_1$ which appeared in the study of strong pseudoconvexity and some more domains biholomorphically related to open and convex domains of $\ell_1$ such as its ball.
We study the cluster value problem for certain Banach algebras of holomorphic functions defined on the unit ball of a complex Banach space X. The main results are for spaces of the form X = C(K).
The main result is that the cluster value problem in separable Banach spaces, for the Banach algebras $A_u$ and $H^{\infty}$, can be reduced to the cluster value problem in those spaces which are $\ell_1$ sums of a sequence of finite dimensional spaces.