Adjacent cross-sections of the commutant of Hilbert space operators
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914592783859712 |
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| author | Kérchy, László |
| author_facet | Kérchy, László |
| contents | Applying the techniques resulting the existence of almost invariant half-spaces, similarity models $\wh T$ can be given for upper triangular operator-matrices $T= \left[\begin{matrix}A&C\\ 0&B\end{matrix}\right]$. The model $\wh T$ is also an operator-matrix, containing two diagonal operators in the general case \cite{ker25}, and the unilateral shift $S$ together with a diagonal operator in the particular case when $A$ is similar to $S$ \cite{ker26}. Well-chosen compressions of operators in the commutant $\{\wh T\}'$ form a linear manifold $\whŁ$ satisfying the condition that every $\wh X\in\wh Ł$ is transformed into $\wh Y$ with $\rank\wh Y\le 2$ by a canonical mapping. Furthermore, a cyclcity property of $\{T\}'$ yields transitivity of $\whŁ$. In \cite{ker25} and \cite{ker26} the 3-dimensional cross-sections of $\whŁ$ have been investigated characterizing the canonical bases occurring in the corresponding subspaces of the matrix-algebra $M_3[\C]$. In this paper new conditions are provided by studying matching of adjacent cross-sections. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23846 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Adjacent cross-sections of the commutant of Hilbert space operators Kérchy, László Functional Analysis Applying the techniques resulting the existence of almost invariant half-spaces, similarity models $\wh T$ can be given for upper triangular operator-matrices $T= \left[\begin{matrix}A&C\\ 0&B\end{matrix}\right]$. The model $\wh T$ is also an operator-matrix, containing two diagonal operators in the general case \cite{ker25}, and the unilateral shift $S$ together with a diagonal operator in the particular case when $A$ is similar to $S$ \cite{ker26}. Well-chosen compressions of operators in the commutant $\{\wh T\}'$ form a linear manifold $\whŁ$ satisfying the condition that every $\wh X\in\wh Ł$ is transformed into $\wh Y$ with $\rank\wh Y\le 2$ by a canonical mapping. Furthermore, a cyclcity property of $\{T\}'$ yields transitivity of $\whŁ$. In \cite{ker25} and \cite{ker26} the 3-dimensional cross-sections of $\whŁ$ have been investigated characterizing the canonical bases occurring in the corresponding subspaces of the matrix-algebra $M_3[\C]$. In this paper new conditions are provided by studying matching of adjacent cross-sections. |
| title | Adjacent cross-sections of the commutant of Hilbert space operators |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2605.23846 |