Article
| Title: | On the range-kernel orthogonality of some elementary operators (English) |
| Author: | El Mouadine, Hamza |
| Author: | Faouzi, Abdelkhalek |
| Author: | Ech-chad, Mohamed |
| Author: | Bouhafsi, Youssef |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 3 |
| Year: | 2026 |
| Pages: | 839-847 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $L(H)$ denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space $H$ and let $(\mathcal {J},\|{\cdot }\|_{\mathcal {J}})$ denote a norm ideal in $L(H)$. For $A,B\in L(H)$, the generalized derivation $\delta _{A,B}$ and the multiplication operator $M_{A,B}$ are defined on $L(H)$ by $\delta _{A,B}(X)=AX-XB$ and $M_{A,B}(X)=AXB$. Denote $\tau _{A,B}=M_{A,B}-I$ and $d_{A,B}=\delta _{A,B}$ or $\tau _{A,B}$. We give some pairs $(A,B)$ of operators $A$ and $B$ such that the range $R(d_{A,B}|\mathcal {J})$ is orthogonal to the kernel $\ker (d_{A,B}|\mathcal {J})$ of the restriction $d_{A,B}|\mathcal {J}$ with respect to unitarily invariant norm $\|{\cdot }\|_{\mathcal {J}}$. It is proved that if $A$ and $B^{\ast }$ are subnormal, then the Hilbert-Schmidt operators in the range of $M_{A,B}$ are orthogonal to those in the kernel of $M_{A,B}$. We also give some related results. (English) |
| Keyword: | generalized derivation |
| Keyword: | elementary operator |
| Keyword: | orthogonality |
| Keyword: | subnormal operator |
| Keyword: | compact operator |
| MSC: | 47A16 |
| MSC: | 47A30 |
| MSC: | 47B07 |
| MSC: | 47B20 |
| MSC: | 47B47 |
| DOI: | 10.21136/CMJ.2026.0374-25 |
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| Date available: | 2026-09-11T13:12:22Z |
| Last updated: | 2026-09-14 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153730 |
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