Angular Derivatives of Holomorphic Maps in Infinite Dimensions

Let H be a complex Hilbert space and let L H, H.be the complex Banach space of all bounded linear operators from H to H with the operator norm. In the generalized right half-planes and the unit balls contained in H and L H, H., infinite-dimensional angular sets are defined, various new generalizations of the classical Pick]Julia theorem to infinite dimensions are proved, and conditions of Carath´eodory]Fan type, guaranteeing the existence of angular limits and angular derivatives of holomorphic maps of these generalized right half-planes and unit balls, are established.