Formal Functional Calculus for Weakly Locally Nilpotent Operators in Fréchet Spaces

We generalize the classical Riesz–Dunford holomorphic functional calculus with weakly locally nilpotent operators in a Fréchet space to the case of the algebra of all formal power series. We obtain a counterpart of the spectral mapping theorem and formulas similar to the Taylor and Riesz–Dunford formulas. We show that the weak local nilpotency of an operator is necessary for constructing the formal functional calculus. Bibliography: 12 titles.