On iterations of non-negative operators and their applications to ellipticsystems

Let H-0, H-1 be Hilbert spaces and L: H-0 - H-1 be a linear bounded operato r with \\L\\ less than or equal to 1 Then L* L is a bounded linear self-adj oint non-negative operator in the Hilbert space Ho and one can use the Neum ann series Sigma (infinity)(nu =o)(I - L* L)L-nu* f in order to study solva bility of the operator equation Lu = f. In particular, applying this method to the ill-posed Cauchy problem for sol utions to an elliptic system Pu = 0 of linear PDE's of order p with smooth coefficients we obtain solvability conditions and representation formulae f or solutions of the problem in Hardy spaces whenever these solutions exist. For the Cauchy-Riemann system in C the summands of the Neumann series are iterations of the Cauchy type integral.