Boundary value matrix problems and Drazin invertible operators

Let $A$ and $B$ be given linear operators on Banach spaces $X$ and $Y$, we denote by $M_C$ the operator defined on $X \oplus Y$ by $M_{C}=\begin{pmatrix}A & C \\ 0 & B%\end{pmatrix}.$In this paper, we study an abstract boundaryvalue matrix problems with a spectral parameter described by Drazin invertibile operators of the form $$\begin{cases}U_L=\lambda M_{C}w+F, & \\\Gamma w=\Phi, & \end{cases}%$$where $U_L , M_C$ are upper triangular operators matrices $(2\times 2)$ acting in Banach spaces, $\Gamma$ is boundary operator, $F$ and $\Phi $ are given vectors and $\lambda $ is a complex spectral parameter.We introduce theconcept of initial boundary operators adapted to the Drazin invertibility andwe present a spectral approach for solving the problem. It can be shown thatthe considered boundary value problems are uniquely solvable and that theirsolutions are explicitly calculated. As an application we give an example to illustrate our results.