On the Abel basis property of the system of root vector-functions of degenerate elliptic differential operators with singular matrix coefficients

We establish completeness and summability in the Abel–Lidskii sense for the system of root vector-functions of nonselfadjoint elliptic matrix operators $A$ generated by noncoercive forms with the Dirichlet-type boundary conditions. An operator $A+\beta E$ is positive for a sufficiently large $\beta>0$ but not strongly positive in general. We obtain estimates for the eigenvalues and resolvent of $A$. Also, we study the resolvent of the extension$\mathscr{A}$ of $A$ to the corresponding negative space.