Unbounded Toeplitz operators and finite rank de Branges-Rovnyak spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916022353657856 |
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| author | Ghara, Soumitra Reza, MD Ramiz Sahu, Chaman Kumar |
| author_facet | Ghara, Soumitra Reza, MD Ramiz Sahu, Chaman Kumar |
| contents | Motivated by the recent developments of de Branges-Rovnyak spaces, we investigate the function theoretic aspects of finite rank de Branges-Rovnyak spaces $H(B)$ generated by row-valued Schur functions $B$. We provide a generalization of Sarason's fundamental work by characterizing finite rank $H(B)$-spaces as the domain of the adjoint of the Toeplitz operators $T_φ^*$ with symbol $φ= BA^{-1}$, where $A$ is an matrix-valued outer function satisfying $A^*A+B^*B = I$ a.e. on the unit circle. We derive a norm formula for functions in $H(B)$-space and provide a concrete realization of this norm in terms of the Taylor coefficients of the function and the symbol $φ$. As an application, we characterize all symbols $B$ for which $H^\infty \subseteq H(B)$ in terms of the boundary behavior of $I-BB^*$, thereby extending Sarason's criterion for the classical de Branges-Rovnyak spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17861 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unbounded Toeplitz operators and finite rank de Branges-Rovnyak spaces Ghara, Soumitra Reza, MD Ramiz Sahu, Chaman Kumar Functional Analysis 30H45, 46E22, 47B35 Motivated by the recent developments of de Branges-Rovnyak spaces, we investigate the function theoretic aspects of finite rank de Branges-Rovnyak spaces $H(B)$ generated by row-valued Schur functions $B$. We provide a generalization of Sarason's fundamental work by characterizing finite rank $H(B)$-spaces as the domain of the adjoint of the Toeplitz operators $T_φ^*$ with symbol $φ= BA^{-1}$, where $A$ is an matrix-valued outer function satisfying $A^*A+B^*B = I$ a.e. on the unit circle. We derive a norm formula for functions in $H(B)$-space and provide a concrete realization of this norm in terms of the Taylor coefficients of the function and the symbol $φ$. As an application, we characterize all symbols $B$ for which $H^\infty \subseteq H(B)$ in terms of the boundary behavior of $I-BB^*$, thereby extending Sarason's criterion for the classical de Branges-Rovnyak spaces. |
| title | Unbounded Toeplitz operators and finite rank de Branges-Rovnyak spaces |
| topic | Functional Analysis 30H45, 46E22, 47B35 |
| url | https://arxiv.org/abs/2605.17861 |