TY - JOUR
T1 - The Boundedness Below of $2×2$ Upper Triangular Linear Relation Matrices
AU - Huo , Ran
AU - Du , Yanyan
AU - Huang , Junjie
JO - Journal of Mathematical Study
VL - 1
SP - 71
EP - 83
PY - 2024
DA - 2024/03
SN - 57
DO - http://doi.org/10.4208/jms.v57n1.24.04
UR - https://global-sci.org/intro/article_detail/jms/22988.html
KW - Linear relation matrix, boundedness below, approximate point spectrum, space decomposition.
AB - <div style="text-align: justify;">In this note, the boundedness below of linear relation matrix $M_{C}=\left(\begin{smallmatrix}&nbsp; A &amp; C \\&nbsp; 0 &amp; B\\&nbsp; \end{smallmatrix} \right)\in LR(H\oplus K)$ 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.</div>