In this note, the boundedness below of linear relation matrix $M_{C}=\left(\begin{array}{cc}A & C \\ 0 & B\end{array}\right) \in L R(H \oplus K)$ is considered, where $A \in C L R(H), B \in C L R(K), C \in B L R(K, H), H, K$ are separable Hilbert spaces. By suitable space decompositions, a necessary and sufficient condition for diagonal relations A,B is given so that $M_{C}$ is bounded below for some $C \in B L R(K, H)$. Besides, the characterization of $\sigma_{a p}\left(M_{C}\right)$ and $\sigma_{s u}\left(M_{C}\right)$ are obtained, and the relationship between $\sigma_{a p}\left(M_{0}\right)$ and $\sigma_{a p}\left(M_{C}\right)$ is further presented.
Acknowledgments
We are very grateful to the anonymous referees for their valuable suggestions and comments which help to improve this paper. This work was supported by the National Natural Science Foundation of China (Grant No. 11961052), the Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region (Grant No. NMGIRT2317), and the Natural Science Foundation of Inner Mongolia (Grant No. 2021MS010- 06). No potential competing interests were reported by the authors.
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