Some properties of upper triangular $$3\times 3$$ 3 × 3 -block matrices of linear relations

Abstract

In this paper, we define the upper triangular \(3\times 3\)-block matrices in the form, $$\begin{aligned} \mathcal { M}_{C}=\left( \begin{array}{ccc} A_{1}&{}\quad B &{}\quad C \\ 0 &{}\quad A_{2} &{}\quad D \\ 0 &{}\quad 0 &{}\quad A_{3} \\ \end{array} \right) . \end{aligned}$$ We define the adjoint of upper triangular \(3\times 3\)-block matrices. Accordingly it is has been proven that if is Fredholm linear relation for some bounded linear operators B, C and D, then \( A_{1},~ A_{2}\) and \(A_{3}\) are Fredholm linear relations. Finally, we undertake a detailed treatment of some subsets of the essential spectra of \( \mathcal {M}_{C}\) (see Theorem 3.2). This study led us to generalize some well-known results for the single valued block matrices (see Ammar et al. in Ann Funct Anal 4(2):153–170, 2013 and A. Jeribi in Spectral Theory and applications of linear operators and block operator matrices. Springer, New York, 2015, Chapter 10).