The numerical ranges of the generalized quadratic operators
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912690585206784 |
|---|---|
| author | Wu, Kangjian Xu, Qingxiang |
| author_facet | Wu, Kangjian Xu, Qingxiang |
| contents | We investigate the generalized quadratic operator defined by $$T =\left(
\begin{array}{cc}
a I_H & A \\ c A^* & bI_K
\end{array}
\right) ,$$ where $H$ and $K$ are Hilbert spaces, $A:K\to H$ is a bounded linear operator, $I_H$ and $I_K$ denote the identity operators on $H$ and $K$, respectively, and $a,b,c$ are complex numbers. It is shown that $T$ attains its norm if and only if $A$ attains its norm. Furthermore, a complete characterization of the numerical range of $T$ is provided by a new approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04313 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The numerical ranges of the generalized quadratic operators Wu, Kangjian Xu, Qingxiang Functional Analysis We investigate the generalized quadratic operator defined by $$T =\left( \begin{array}{cc} a I_H & A \\ c A^* & bI_K \end{array} \right) ,$$ where $H$ and $K$ are Hilbert spaces, $A:K\to H$ is a bounded linear operator, $I_H$ and $I_K$ denote the identity operators on $H$ and $K$, respectively, and $a,b,c$ are complex numbers. It is shown that $T$ attains its norm if and only if $A$ attains its norm. Furthermore, a complete characterization of the numerical range of $T$ is provided by a new approach. |
| title | The numerical ranges of the generalized quadratic operators |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2511.04313 |