Boundary problems for Helmholtz equation and the Cauchy problem for Dirac operators

Studying an operator equation $Au=f$ in Hilbert spaces one usually needs the adjoint operator $A^\star$ for $A$. Solving the ill-posed Cauchy problem for Dirac type systems in the Lebesgue spaces by an iteration method we propose to construct the corresponding adjoint operator with the use of normally solvable mixed problem for Helmholtz Equation. This leads to the description of necessary and sufficient solvability conditions for the Cauchy Problem and formulae for its exact and approximate solutions.