Properties of the Brunel Operator

For T:X->X a Cesàro bounded linear operator on a Banach space X and for ψ the function on the closed unit disk given by ψ(z)=(1-sqrt(1-z))/z, we show that the Brunel operator A(T)=ψ(T) is power-bounded and satisfies the Ritt condition ||A^(n+1)(T)-A^n(T)||=O(1/n). We prove that when X is reflexive and T is mean ergodic, the invariant subspaces and coboundaries of T and A(T) coincide, as well as that the powers of the Brunel operator converge in the strong operator topology to the same T-invariant limit as the Cesàro averages of T. We show that when T is positive, the norm (resp. pointwise) convergence of the Cesàro averages of T is equivalent to the norm (resp. pointwise) convergence of the powers of A(T); in particular, this reduces the study of the pointwise convergence of the Cesàro averages of positive Cesàro bounded operators to the study of the pointwise convergence of the powers of positive Ritt operators. Alongside these results, we give an analogue of the spectral mapping theorem for the Brunel operator, and derive several asymptotic estimates for the series coefficients of the resolvent of A(T). Finally, we conclude by analyzing the iterates A(T), A(A(T)), ..., and give sufficient conditions for the limit of this iteration process to have the same invariant subspace and coboundary as T.Bachelor of Scienc