# Boundary value matrix problems and Drazin invertible operators

### Abstract

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 boundary

value 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 the

concept of initial boundary operators adapted to the Drazin invertibility and

we present a spectral approach for solving the problem. It can be shown that

the considered boundary value problems are uniquely solvable and that their

solutions are explicitly calculated. As an application we give an example to illustrate our results.

### References

F.V. Atkinson, Disrete and continuous boundary problems, Academic Press, New York, 1964.

F.V. Atkinson, H. Langer, R. Mennicken, A.A. Shkalikov, The essential spectrum of some matrix operators, Math. Nachr., 167 (1994), 5–20.

J. Behrndt, M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal., 243 (2007), 536–565.

P.L. Butzer, J.J. Koliha, The a-Drazin inverse and ergodic behaviour of semigroups and cosine operator functions, J. Operator Theory, 62 (2009), 297—326.

S. Hassi, H. de Snoo, F. Szafraniec, Operator methods for boundary value problems, Cambridge University Press, 2012.

N. Khaldi, M. Benharrat, B. Messirdi, On the spectral boundary value problems and boundary approximate controllability of linear systems, Rend. Circ. Mat. Palermo, 63 (2014), 141—153.

N. Khaldi, M. Benharrat, B. Messirdi, Linear boundary-value problems described by Drazin invertible operators, Math. Notes, 101 (2017), 994–999.

J.J. Koliha, T.D. Tran, The Drazin inverse for closed linear operators and the asymptotic convergence of C0 -semigroups, J. of Operator Theory, 46 (2001), 323–336.

A.M. Krall, Hilbert spaces, Boundary value problems and orthogonal polynomials, Springer, 2002.

P.J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics 107, Second Edition, Springer-Verlag, New York, 1993.

C. Tretter, Spectral theory of block operator matrices and applications, Imperial College Press, London, 2008.

*Matematychni Studii*,

*57*(1), 16-22. https://doi.org/10.30970/ms.57.1.16-22

Copyright (c) 2022 K. Miloud Hocine

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.