A Characterization of the Essential Spectra of $$2\times 2$$ 2 × 2 Block Matrices of Linear Relations

The main goal of this paper is to elaborate some results on the spectral properties of $$2 \times 2$$ block matrix linear relations with unbounded entries and with a domain consisting of vectors which satisfy certain relations between their components. We present some conditions to prove some Frobenius–Schur decompositions for linear relations and characterize the stability of the essential spectra of these linear relations. Our results generalize many known ones in the literature, in particular those obtained by Alvarez et al. (in Math Methods Appl Sci 37:620–644, 2014) and Batkai et al. (in Math Nachr 278:1408–1429, 2005).