Spectral property of upper triangular relation matrices

ABSTRACT

Let H and K be infinite dimensional separable Hilbert spaces. For $A\in \mathcal{L}\mathcal{R}\left(H\right)$, $B\in \mathcal{L}\mathcal{R}\left(K\right)$ and $C\in \mathcal{L}\mathcal{R}(K,H)$, we denote by $M_C=\left(\begin{array}{cc}A& C\\ 0& B\end{array}\right)$ the upper triangular relation matrix. In this paper, the sets $\bigcap _{C\in {\mathcal{C}}_M(K,H)}\sigma \left(M_C\right)$, $\bigcap _{C\in \mathcal{L}\mathcal{R}(K,H)}{\sigma }_1\left(M_C\right)$ and $\bigcap _{C\in \mathcal{B}(K,H)}{\sigma }_2\left(M_C\right)$ are characterized, where ${\sigma }_1\in \{{\sigma }_p,{\sigma }_r,{\sigma }_c\}$ and ${\sigma }_2\in \{\sigma ,{\sigma }_p,{\sigma }_r,{\sigma }_c\}$. Moreover, the relationship between $\sigma \left(M_C\right)$, $\sigma \left(A\right)$ and $\sigma \left(B\right)$ is described for given relations $A\in \mathcal{B}\mathcal{C}\mathcal{R}\left(H\right)$, $B\in \mathcal{B}\mathcal{C}\mathcal{R}\left(K\right)$ and $C\in \mathcal{B}\mathcal{R}(K,H)$ under the local spectral theory.

Keywords:

• Hilbert spaces
• relation matrix
• spectrum

2010 Mathematics Subject Classifications:

• 47A06
• 47A10
• 47A55

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable and constructive suggestions and comments which have greatly improved this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the NNSF of China [grant number 11961052]), the Research Program of Sciences at Universities of Inner Mongolia (No. NZJJ18018), and the Postgraduate Scientific Research Innovation Foundation of Inner Mongolia University [grant number 10000-15010109].