On the chain of commuting operators on Banach spaces and Lomonosov's invariant subspace theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913948845998080 |
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| author | Szczepanski, Tomasz |
| author_facet | Szczepanski, Tomasz |
| contents | An operator $T$ on a Banach space is said to be of chain $N$ if there exist non-scalar operators $S_1,...,S_{N-1}$ and a non-zero compact $K$ such that $$T \leftrightarrow S_1 \leftrightarrow S_2 \leftrightarrow ...\leftrightarrow S_{N-1} \leftrightarrow K,$$ where $A\leftrightarrow B$ means $AB=BA$. A connection of this theory to the Lomonosov's Invariant Subspace Theorem is highlighted. It is shown that for every weighted shift $T$ it is of chain $3$. In particular, every non-Lomonosov operator from from the work of Hadwin et al. is of chain $3$. An example of an operator on a separable Hilbert space is given, such that it fails to be connected to a compact operator via a chain of any length. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_14297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the chain of commuting operators on Banach spaces and Lomonosov's invariant subspace theorem Szczepanski, Tomasz Functional Analysis Operator Algebras 47A65 (Primary) 47A15 (Secondary) An operator $T$ on a Banach space is said to be of chain $N$ if there exist non-scalar operators $S_1,...,S_{N-1}$ and a non-zero compact $K$ such that $$T \leftrightarrow S_1 \leftrightarrow S_2 \leftrightarrow ...\leftrightarrow S_{N-1} \leftrightarrow K,$$ where $A\leftrightarrow B$ means $AB=BA$. A connection of this theory to the Lomonosov's Invariant Subspace Theorem is highlighted. It is shown that for every weighted shift $T$ it is of chain $3$. In particular, every non-Lomonosov operator from from the work of Hadwin et al. is of chain $3$. An example of an operator on a separable Hilbert space is given, such that it fails to be connected to a compact operator via a chain of any length. |
| title | On the chain of commuting operators on Banach spaces and Lomonosov's invariant subspace theorem |
| topic | Functional Analysis Operator Algebras 47A65 (Primary) 47A15 (Secondary) |
| url | https://arxiv.org/abs/2507.14297 |