On similarity to contractions of class $C_{\cdot 0}$ with finite defects
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909780517322752 |
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| author | Gamal', Maria F. |
| author_facet | Gamal', Maria F. |
| contents | A criterion on the similarity of a (bounded, linear) operator $T$ on a (complex, separable) Hilbert space $\mathcal H$ in terms of shift-type invariant subspaces of $T$ to a contraction of class $C_{\cdot 0}$ with finite unequal defects is given. Namely, $T$ is similar to such a contraction if and only if the minimal quantity of (closed) invariant subspaces $\mathcal M$ of $T$ such that the restriction $T|_{\mathcal M}$ of $T$ on $\mathcal M$ is similar to the simple unilateral shift, whose linear span is $\mathcal H$, is finite. A sufficient condition for the similarity of an absolutely continuous polynomially bounded operator $T$ to a contraction of class $C_{\cdot 0}$ with finite equal defects is given. Namely, $T$ is similar to such a contraction if the (spectral) multiplicity of $T$ is finite and $B(T)=\mathbb O$, where $B$ is a finite product of Blaschke products with simple zeros satisfying the Carleson interpolating condition (a Carleson--Newman product). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_15016 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On similarity to contractions of class $C_{\cdot 0}$ with finite defects Gamal', Maria F. Functional Analysis 47A45, 47A15 A criterion on the similarity of a (bounded, linear) operator $T$ on a (complex, separable) Hilbert space $\mathcal H$ in terms of shift-type invariant subspaces of $T$ to a contraction of class $C_{\cdot 0}$ with finite unequal defects is given. Namely, $T$ is similar to such a contraction if and only if the minimal quantity of (closed) invariant subspaces $\mathcal M$ of $T$ such that the restriction $T|_{\mathcal M}$ of $T$ on $\mathcal M$ is similar to the simple unilateral shift, whose linear span is $\mathcal H$, is finite. A sufficient condition for the similarity of an absolutely continuous polynomially bounded operator $T$ to a contraction of class $C_{\cdot 0}$ with finite equal defects is given. Namely, $T$ is similar to such a contraction if the (spectral) multiplicity of $T$ is finite and $B(T)=\mathbb O$, where $B$ is a finite product of Blaschke products with simple zeros satisfying the Carleson interpolating condition (a Carleson--Newman product). |
| title | On similarity to contractions of class $C_{\cdot 0}$ with finite defects |
| topic | Functional Analysis 47A45, 47A15 |
| url | https://arxiv.org/abs/2405.15016 |