| Title:
|
Some Berezin number inequalities for operator matrices (English) |
| Author:
|
Bakherad, Mojtaba |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
4 |
| Year:
|
2018 |
| Pages:
|
997-1009 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The Berezin symbol $\tilde {A}$ of an operator $A$ acting on the reproducing kernel Hilbert space ${\mathcal H}={\mathcal H}(\Omega )$ over some (nonempty) set is defined by $\tilde {A}(\lambda )=\langle A\hat {k}_{\lambda },\hat {k}_{\lambda }\rangle ,$ $\lambda \in \Omega $, where $\hat {k}_{\lambda }={{k}_{\lambda }}/{\|{k}_{\lambda }\|}$ is the normalized reproducing kernel of ${\mathcal H}$. The Berezin number of the operator $A$ is defined by ${\bf ber}(A)=\sup _{\lambda \in \Omega }|\tilde {A}(\lambda )|=\sup _{\lambda \in \Omega }|\langle A\hat {k}_{\lambda },\hat {k}_{\lambda }\rangle |$. Moreover, ${\bf ber}(A)\leq w(A)$ (numerical radius). We present some Berezin number inequalities. Among other inequalities, it is shown that if ${\bf T}=\left [\smallmatrix A&B\\ C&D \endmatrix \right ]\in {\mathbb B}({\mathcal H(\Omega _1)}\oplus {\mathcal H(\Omega _2)})$, then $$ {\bf ber}({\bf T}) \leq \frac {1}{2}({\bf ber}(A)+{\bf ber}(D))+\frac {1}{2}\sqrt {({\bf ber}(A)- {\bf ber}(D))^2+(\|B\|+\|C\|)^2}. $$ (English) |
| Keyword:
|
reproducing kernel |
| Keyword:
|
Berezin number |
| Keyword:
|
numerical radius |
| Keyword:
|
operator matrix |
| MSC:
|
15A60 |
| MSC:
|
30E20 |
| MSC:
|
47A12 |
| MSC:
|
47A30 |
| MSC:
|
47B15 |
| MSC:
|
47B20 |
| idZBL:
|
Zbl 07031692 |
| idMR:
|
MR3881891 |
| DOI:
|
10.21136/CMJ.2018.0048-17 |
| . |
| Date available:
|
2018-12-07T06:19:16Z |
| Last updated:
|
2021-01-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147516 |
| . |
| Reference:
|
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| Reference:
|
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