Symmetric functions can be considered as operators acting on the ring of polynomials with coeecients in R. We present the package SFA, an implementation of this action for the computer algebra system Maple. As an example, we show how to recover diierent classical expressions of Lagrange inversion, and of Faber polynomials.

Let A be a bounded linear operator in a Hilbert space H with spectrum Λ(A). The Kreiss matrix theorem gives bounds based on the resolvent norm ‖(zI−A)−1‖ for ‖An‖ if Λ(A) is in the unit disk or for ‖etA‖ if Λ(A) is in the left half-plane. We generalize these results to a complex domain Ω, giving bounds for ‖Fn(A)‖ if Λ(A) ⊂ Ω, where Fn denotes the nth Faber polynomial associated with Ω. One of ...