The Boundedness Below of $2×2$ Upper Triangular Linear Relation Matrices
Ran Huo, Yanyan Du, Junjie Huang · Journal of Mathematical Study · 2024
In this note, the boundedness below of linear relation matrix $M_{C}=\left(\begin{smallmatrix} A & C \ 0 & B\ \end{smallmatrix} \right)\in LR(H\oplus K)$ is considered, where $A\in CLR(H)$, $B\in CLR(K),$ $C\in BLR(K,H)$, $H,K$ 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 $C\in BLR(K,H)$. Besides, the characterization of $\sigma_{ap}(M_{C})$ and $\sigma_{su}(M_{C})$ are obtained, and the relationship between $\sigma_{ap}(M_{0})$ and $\sigma_{ap}(M_{C})$ is further presented.