In this note we summarize a holomorphic vector function and a holomorphic operator function in a complex Banach space X. By judicious mixture of the classic Cauchy formula, functional resorent equation and spectral projection, we introduce a theorem t...
In this note we summarize a holomorphic vector function and a holomorphic operator function in a complex Banach space X. By judicious mixture of the classic Cauchy formula, functional resorent equation and spectral projection, we introduce a theorem that a bounded linear operator in the form of Cauchy's integral is a holomorphic map of the algebra of function into the set of all operators of a complex Banach space into itself.