Linear relation matrix, boundedness below, approximate point spectrum, space decomposition.
Abstract
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.
