Convergence of the Neumann series in higher norms

Natural conditions on an operator A are given so that the Neumann series for (Id+A) converges in higher norm topologies. 1 The general case In applications one often considers equations of the form (Id+A)u = v (1) where A is a bounded linear operator. In this note we consider the convergence of the Neumann series solution for this type of equation. We first prove a fairly abstract result and then consider a variety of special cases. Suppose that {Xk} is a nested collection of Banach spaces, X0 ⊃ X1 ⊃ . . . Xk ⊃ Xk+1 ⊃ . . . , with norms {‖ · ‖k}. Let |||A||| denote the operator norm of A : X0 → X0, |||A||| = sup X03u6=0 ‖Au‖0 ‖u‖0 . (2) If |||A||| < α0 < 1 then it is well known that the Neumann series for (Id+A) converges, see [1]. Indeed, if we define the sequence u0 = v uj = v − Auj−1 (3) then uj is the jth partial sum of the Neumann series. A simple induction argument shows that ‖uj+1 − uj‖0 ≤ α 0‖u1 − u0‖0. (4) Research partially supported by NSF grant DMS02–03705 and the Francis J. Carey chair. Address: Department of Mathematics, University of Pennsylvania, Philadelphia, PA. E-mail: [email protected]