ABSTRACT
The present chapter presents some problems related to the maximum principle for holomorphic vector-valued functions defined on domains of the complex plane. The fundamental role of holomorphic functions in the theory of linear operators is now well known. It is worth mentioning that the use of holomorphic functions in the theory of linear operators goes back to H. Poincaré, Fr. Riesz, and G. Giorgi. Important contributions to the theory of holomorphic (analytic) functions with values in Banach spaces were given by N. Wiener, L. Fantappié, N. Dunford, I. M. Gelfand, A. E. Taylor, and many others. For a detailed account of this theory we refer to the books of E. Hille and R. S. Phillips (1957), and N. Dunford and J. Schwartz (1958).
