A formula of $A$-spectral radius for $A^{\frac{1}{2}}$-adjoint operators on semi-Hilbertian spaces
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913237935587328 |
|---|---|
| author | Majumdar, Arup Johnson, P. Sam |
| author_facet | Majumdar, Arup Johnson, P. Sam |
| contents | In this paper, we prove the relation $\frac{r_{A}(T) + r_{A}(T^{\diamond}) + |r_{A}(T^{\diamond}) - r_{A}(T)|}{2} = \sup \{ |λ|: λ\in σ_{A}(T)\}$, where $A$ is a positive semidefinite operator (not necessarily to have a closed range) and $r_{A}(T)$ is the $A$-spectral radius of $T$ in $B_{A^{\frac{1}{2}}}(H)$. Also we prove that $\sup \{ |λ|: λ\in σ_{A}(T)\} = r_{A}(T), \text{ when } T \in B_{A^{\frac{1}{2}}}(H) \text { commutes with } A$. By introducing $A$-Harte spectrum $σ_{A_{h}}(\mathbf{T})$ of a $d$-tuple operator $\mathbf{T}= (T_{1},\dots,T_{d}) \in (B_{A^{\frac{1}{2}}}(H))^{d}$, we prove that $r_{A_{h}}(\mathbf{T}) \leq \sup \{\|λ\|_{2}: λ\in σ_{A_{h}}(\mathbf{T})\}$, where $r_{A_{h}}(\mathbf{T})$ is the $A$-Harte spectral radius of $\mathbf{T}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12961 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A formula of $A$-spectral radius for $A^{\frac{1}{2}}$-adjoint operators on semi-Hilbertian spaces Majumdar, Arup Johnson, P. Sam Functional Analysis 47A10, 46C05, 47A30, 47A80 In this paper, we prove the relation $\frac{r_{A}(T) + r_{A}(T^{\diamond}) + |r_{A}(T^{\diamond}) - r_{A}(T)|}{2} = \sup \{ |λ|: λ\in σ_{A}(T)\}$, where $A$ is a positive semidefinite operator (not necessarily to have a closed range) and $r_{A}(T)$ is the $A$-spectral radius of $T$ in $B_{A^{\frac{1}{2}}}(H)$. Also we prove that $\sup \{ |λ|: λ\in σ_{A}(T)\} = r_{A}(T), \text{ when } T \in B_{A^{\frac{1}{2}}}(H) \text { commutes with } A$. By introducing $A$-Harte spectrum $σ_{A_{h}}(\mathbf{T})$ of a $d$-tuple operator $\mathbf{T}= (T_{1},\dots,T_{d}) \in (B_{A^{\frac{1}{2}}}(H))^{d}$, we prove that $r_{A_{h}}(\mathbf{T}) \leq \sup \{\|λ\|_{2}: λ\in σ_{A_{h}}(\mathbf{T})\}$, where $r_{A_{h}}(\mathbf{T})$ is the $A$-Harte spectral radius of $\mathbf{T}$. |
| title | A formula of $A$-spectral radius for $A^{\frac{1}{2}}$-adjoint operators on semi-Hilbertian spaces |
| topic | Functional Analysis 47A10, 46C05, 47A30, 47A80 |
| url | https://arxiv.org/abs/2402.12961 |