Aiena's local spectral theory for a block matrix linear relations through localized SVEP

The issue of the single-valued extension property (abbreviated SVEP) for linear operators introduced by Aiena (Fredholm and local spectral theory, with applications to multipliers, Kluwer Academic Publishers, Dordrecht, 2004) and (Fredholm and local spectral theory II. With application to Weyl-type theorems, Springer, Cham, 2018), motivates several authors to develope this notion for a block linear operators matrices. In this paper, enlightened by the study of Ammar et al. (Mediterr J Math 18(2):1-27, 2021), we investigate a few properties of the local spectra of a 2 x 2 block matrix of linear relations. Besides, by the new sets originating from the SVEP, we give the necessary and sufficient conditions to characterize the spectra and the essential spectra.