Perturbations of Weyl spectra for 2 × 2 relation matrices

This paper solves the completion problem for 2 × 2 block operator matrices whose entries are linear relations. We provide necessary and sufficient conditions for the completed matrix Mx to be Weyl, right Weyl, or left Weyl, whether the unknown entry X is a single-valued bounded operator or a multi-valued bounded relation. Consequently, we characterize the perturbations of the associated Weyl spectra. Our results generalize known theorems for operator matrices and provide tools for analyzing systems beyond the scope of standard operator theory.