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

Let H₀, H₁ be Hilbert spaces and L : H₀ → H₁ be a linear bounded operator with ∥L∥ ≤ 1. Then L*L is a bounded linear self–adjoint non–negative operator in the Hilbert space H₀ 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.