Perturbations of Weyl spectra for $2\times 2$ relation matrices

Abstract

This paper solves the completion problem for $2\times 2$ block operator matrices whose entries
are linear relations. We provide necessary and sufficient conditions for the completed
matrix $M_{X}$ 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.