On the solvability of a class of operator differential equations of the second order on the real axis

Solvability of the operator dierential equation of the second order with variable coefficients on the real axis in a certain weight space is studied. The main part of the equation is an abstract elliptic equation in Hilbert space. We note that sufficient conditions on operator coefficients of the perturbed part, preserving ellipticity of the equation, are found in the paper, and estimations of the norms of intermediate derivative operators via the main part of the operator differential equation in a certain weight space are also obtained.