Marchenko V. On spectral basis properties of operators of evolution equations

The conditions of the well/ill-posedness of evolution equations with operators, which have eigenvalues clustering at the infinity along the imaginary axis and eigenvectors not forming a Schauder basis, depending on the character of the behavior of eigenvalues at the infinity, were obtained. The notion of the symmertically-spectral operator was introduced and the conditions of the well-posedness for equations with such operators in l_p and c_0 spaces were obtained. The explicit form of solutions for well-posed equations was found, their asymptotic behavior was studied. We extend the theorem of T. Kato to the case of Schauder decompositions in some Banach spaces. The results were applied to the question of stability of evolution equations.