Density of irreducible operators in the trace-class norm
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915937389641728 |
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| author | Fang, Junsheng Jiang, Chunlan Ma, Minghui Shen, Junhao Shi, Rui Wang, Tianze |
| author_facet | Fang, Junsheng Jiang, Chunlan Ma, Minghui Shen, Junhao Shi, Rui Wang, Tianze |
| contents | In 1968, Paul Halmos initiated the research on density of the set of irreducible operators on a separable Hilbert space. Through the research, a long-standing unsolved problem inquires: is the set of irreducible operators dense in $B(H)$ with respect to the trace-class norm topology? Precisely, for each operator $T $ in $B(H)$ and every $\varepsilon >0$, is there a trace-class operator $K$ such that $T+K$ is irreducible and $\Vert K \Vert_1 < \varepsilon$?
For $p>1$, to prove the $\Vert \cdot \Vert_p$-norm density of irreducible operators in $B(H)$, a type of Weyl-von Neumann theorem effects as a key technique. But the traditional method fails for the case $p=1$, where by $\Vert \cdot \Vert_p$-norm we denote the Schatten $p$-norm.
In the current paper, for a large family of operators in $B(H)$, we give the above long-term problem an affirmative answer. The result is derived from a combination of techniques in both operator theory and operator algebras. Moreover, we discover that there is a strong connection between the problem and another related operator-theoretical problem related to type $\mathrm{II}_1$ von Neumann algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17190 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Density of irreducible operators in the trace-class norm Fang, Junsheng Jiang, Chunlan Ma, Minghui Shen, Junhao Shi, Rui Wang, Tianze Operator Algebras Functional Analysis 46L10, 47C15 In 1968, Paul Halmos initiated the research on density of the set of irreducible operators on a separable Hilbert space. Through the research, a long-standing unsolved problem inquires: is the set of irreducible operators dense in $B(H)$ with respect to the trace-class norm topology? Precisely, for each operator $T $ in $B(H)$ and every $\varepsilon >0$, is there a trace-class operator $K$ such that $T+K$ is irreducible and $\Vert K \Vert_1 < \varepsilon$? For $p>1$, to prove the $\Vert \cdot \Vert_p$-norm density of irreducible operators in $B(H)$, a type of Weyl-von Neumann theorem effects as a key technique. But the traditional method fails for the case $p=1$, where by $\Vert \cdot \Vert_p$-norm we denote the Schatten $p$-norm. In the current paper, for a large family of operators in $B(H)$, we give the above long-term problem an affirmative answer. The result is derived from a combination of techniques in both operator theory and operator algebras. Moreover, we discover that there is a strong connection between the problem and another related operator-theoretical problem related to type $\mathrm{II}_1$ von Neumann algebras. |
| title | Density of irreducible operators in the trace-class norm |
| topic | Operator Algebras Functional Analysis 46L10, 47C15 |
| url | https://arxiv.org/abs/2504.17190 |