Cluster sets of analytic functions on Banach spaces

ARON R., CARANDO D., GAMELIN T.W., LASSALLE S., MAESTRE M.

Guanajuato

Congreso; Mathematical Congress of the Americas; 2013

We present joint work with Richard Aron, Daniel Carando, Ted Gamelinand Manuel Maestre. Let $B$ be the open unit ball of a complexBanach space $X$ and $\bar B^{**}$ the closed unit ball of $X^{**}$, thebidual of $X$. Fixed  $f$ a holomorphic function bounded on$B$ and $z\in \bar B^{**}$, the cluster set of $f$ at $z$,$Cl_r(f,z)$, is defined to be the set of all limits of values of $f$along nets in $B$ converging weak-star to $z$. We study uniform algebras of bounded analytic functions on $B$ inrelation to cluster sets. We prove several cluster value theorems,relating cluster sets of a function to its range on the fibers ofthe spectrum of the algebra. These lead to weak versions of thecorona theorem for $\ell_2$ and for $c_0$. We also address the analogous situation for $H_b(X)$, the Fr\'echetalgebra of all entire functions which are bounded on bounded sets of$X$.