Vector and operator-valued holomorphic functions representable by Carleman type formulas

Let be a simply connected domain and let M be a connected subset of its boundary of positive Lebesque measure. With X we denote a separable Hilbert space or the space of bounded linear functionals on . We set f to be an X-valued holomorphic function, and with we denote the class of X-valued holomorphic functions on which belong to the Hardy class near the set M. In our main result, we show that if f belongs to , then f is representable by a Carleman type formula, and conversely, if f is representable by a Carleman type formula, and in some sense has an analytic continuation across M, then f belongs to . Furthermore we show that in general .