Strengthening of spectral radius, numerical radius, and Berezin radius inequalities
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916642651373568 |
|---|---|
| author | Bhunia, Pintu |
| author_facet | Bhunia, Pintu |
| contents | Suppose $\mathcal{H}_1, \mathcal{H}_2, \ldots, \mathcal{H}_n$ are arbitrary complex Hilbert spaces, and ${\bf A}=[A_{ij}]$ is an $n\times n$ operator matrix with $A_{ij}\in \mathcal{B}(\mathcal{H}_j, \mathcal{H}_i).$ We show that $w({\bf A}) \leq w\left(\begin{bmatrix}
a_{ij}
\end{bmatrix}_{i,j=1}^n \right),$ where $w(\cdot)$ denotes the numerical radius and the entries $$ a_{ij}=\begin{cases}
w(A_{ii}) & \textit{if $i=j$},
\sqrt{ \left( \|A_{ij}\|+\|A_{ji}\| \right)^2- \left(\|A_{ij}\| \|A_{ji}\|-w(A_{ji}A_{ij}) \right)}^{} & \textit{if $i<j$},
0 & \textit{if $i>j$.} \end{cases}$$ This bound improves $w({\bf A}) \leq w\left(\begin{bmatrix}
a'_{ij}
\end{bmatrix}_{i,j=1}^n \right),$ where $a'_{ij}=w(A_{ii})$ if $i=j$ and $a'_{ij}=\|A_{ij}\|$ if $i\neq j$.
We deduce an upper bound for the Kronecker products $A\otimes B$, where $A\in \mathcal{M}_n(\mathbb{C})$ and $B\in \mathcal{B}(\mathcal{H}_1)$, which refines Holbrook's classical bound $w(A\otimes B)\leq w(A)\|B\|$, when all entries of $A$ are non-negative. Further, we obtain the Berezin radius inequalities for $n\times n$ operator matrices where the entries are reproducing kernel Hilbert space operators. We provide an example, which illustrates these inequalities for some concrete operators on the Hardy--Hilbert space. Applying the numerical radius bounds, we show that if $A_i \in \mathcal{B}(\mathcal{H}_i, \mathcal{H}_1) $ and $B_i\in \mathcal{B}(\mathcal{H}_1, \mathcal{H}_i)$ for $i=1,2,$ then
\begin{eqnarray*}
r(A_1B_1+A_2B_2) \leq \frac{ 1 }{2 } \left(w(B_1A_1)+w(B_2A_2) \right)
+ \frac{ 1 }{2 } \sqrt{ \left(w(B_1A_1)-w(B_2A_2)\right)^2 + 3\|B_1A_2\|\|B_2A_1\| + η},
\end{eqnarray*} where $η=w(B_2A_1 B_1A_2)$, and $r(\cdot)$ denotes the spectral radius. We also achieve a bound for the roots of an algebraic equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02615 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strengthening of spectral radius, numerical radius, and Berezin radius inequalities Bhunia, Pintu Functional Analysis 47A12, 47A30, 15A60, 47A10, 47B15, 26C10 Suppose $\mathcal{H}_1, \mathcal{H}_2, \ldots, \mathcal{H}_n$ are arbitrary complex Hilbert spaces, and ${\bf A}=[A_{ij}]$ is an $n\times n$ operator matrix with $A_{ij}\in \mathcal{B}(\mathcal{H}_j, \mathcal{H}_i).$ We show that $w({\bf A}) \leq w\left(\begin{bmatrix} a_{ij} \end{bmatrix}_{i,j=1}^n \right),$ where $w(\cdot)$ denotes the numerical radius and the entries $$ a_{ij}=\begin{cases} w(A_{ii}) & \textit{if $i=j$}, \sqrt{ \left( \|A_{ij}\|+\|A_{ji}\| \right)^2- \left(\|A_{ij}\| \|A_{ji}\|-w(A_{ji}A_{ij}) \right)}^{} & \textit{if $i<j$}, 0 & \textit{if $i>j$.} \end{cases}$$ This bound improves $w({\bf A}) \leq w\left(\begin{bmatrix} a'_{ij} \end{bmatrix}_{i,j=1}^n \right),$ where $a'_{ij}=w(A_{ii})$ if $i=j$ and $a'_{ij}=\|A_{ij}\|$ if $i\neq j$. We deduce an upper bound for the Kronecker products $A\otimes B$, where $A\in \mathcal{M}_n(\mathbb{C})$ and $B\in \mathcal{B}(\mathcal{H}_1)$, which refines Holbrook's classical bound $w(A\otimes B)\leq w(A)\|B\|$, when all entries of $A$ are non-negative. Further, we obtain the Berezin radius inequalities for $n\times n$ operator matrices where the entries are reproducing kernel Hilbert space operators. We provide an example, which illustrates these inequalities for some concrete operators on the Hardy--Hilbert space. Applying the numerical radius bounds, we show that if $A_i \in \mathcal{B}(\mathcal{H}_i, \mathcal{H}_1) $ and $B_i\in \mathcal{B}(\mathcal{H}_1, \mathcal{H}_i)$ for $i=1,2,$ then \begin{eqnarray*} r(A_1B_1+A_2B_2) \leq \frac{ 1 }{2 } \left(w(B_1A_1)+w(B_2A_2) \right) + \frac{ 1 }{2 } \sqrt{ \left(w(B_1A_1)-w(B_2A_2)\right)^2 + 3\|B_1A_2\|\|B_2A_1\| + η}, \end{eqnarray*} where $η=w(B_2A_1 B_1A_2)$, and $r(\cdot)$ denotes the spectral radius. We also achieve a bound for the roots of an algebraic equation. |
| title | Strengthening of spectral radius, numerical radius, and Berezin radius inequalities |
| topic | Functional Analysis 47A12, 47A30, 15A60, 47A10, 47B15, 26C10 |
| url | https://arxiv.org/abs/2503.02615 |