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The Boundedness Below of $2×2$ Upper Triangular Linear Relation Matrices

The Boundedness Below of $2×2$ Upper Triangular Linear Relation Matrices
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Year:    2024

Author:    Ran Huo, Yanyan Du, Junjie Huang

Journal of Mathematical Study, Vol. 57 (2024), Iss. 1 : pp. 71–83

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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jms.v57n1.24.04

Journal of Mathematical Study, Vol. 57 (2024), Iss. 1 : pp. 71–83

Published online:    2024-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    13

Keywords:    Linear relation matrix boundedness below approximate point spectrum space decomposition.

Author Details

Ran Huo

Yanyan Du

Junjie Huang