On Iterations of Non–Negative Operators and Their Applications to Elliptic Systems

Let H 0 , H 1 be Hilbert spaces and L : H 0 → H 1 be a linear bounded operator with ∥ L ∥ ≤ 1. Then L * L is a bounded linear self–adjoint non–negative operator in the Hilbert space H 0 and one can use the Neumann series Σ ∞ v =0 ( I — L * L ) v L * f in order to stud solvabilit of the operator equation Lu = f . In particular, applying this method to the ill–posed Cauch problem for solutions to an elliptic system Pu = 0 of linear PDE's of order p with smoothcoefficients we obtain solvabilit conditions and representation formulae for solutions of the problem in Hardy spaces whenever these solutions exist. For the Cauch–Riemann system in ℂ the summands of the Neumann series are iterations of the Cauch type integral.