Weak-Star Continuous Analytic Functions

Canadian journal of mathematics, Tome 47 (1995) no. 4, pp. 673-683

Résumé

Let X be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of X**. If X* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of X*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming X* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with X**.

Détail

DOI : 10.4153/CJM-1995-035-1

Mots-clés : 46G20, 46E15, 46J15, 46J20

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