On the minimum modulus of dual truncated Toeplitz operators
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866918343281213440 |
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| author | Bhuia, Sudip Ranjan Golla, Ramesh Nag, Puspendu |
| author_facet | Bhuia, Sudip Ranjan Golla, Ramesh Nag, Puspendu |
| contents | This article provides a systematic investigation of the minimum modulus of dual truncated Toeplitz operators (DTTOs) $D_φ$ acting on the orthogonal complement of the model space $\mathcal{K}_u^{\perp}$, where $u$ is a nonconstant inner function and $φ\in L^\infty(\T)$. We first establish an explicit formula for the minimum modulus of the compressed shift $S_u$ and its dual $D_u$ in terms of $|u(0)|$, and prove that the minimum is always attained. For normal DTTOs, we derive sharp spectral bounds utilizing the essential range of the symbol and characterize the conditions under which $m(D_φ)$ coincides with the essential infimum of $|φ|$. In the general setting, for unimodular $\vp$, we obtain exact formulas and two sided estimates for $m(D_φ)$ by analyzing the norms of associated Toeplitz and Hankel operators restricted to the model space. Finally, we provide several concrete examples to illustrate our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15713 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the minimum modulus of dual truncated Toeplitz operators Bhuia, Sudip Ranjan Golla, Ramesh Nag, Puspendu Functional Analysis Spectral Theory 47B35, 47A30, and 47B32 This article provides a systematic investigation of the minimum modulus of dual truncated Toeplitz operators (DTTOs) $D_φ$ acting on the orthogonal complement of the model space $\mathcal{K}_u^{\perp}$, where $u$ is a nonconstant inner function and $φ\in L^\infty(\T)$. We first establish an explicit formula for the minimum modulus of the compressed shift $S_u$ and its dual $D_u$ in terms of $|u(0)|$, and prove that the minimum is always attained. For normal DTTOs, we derive sharp spectral bounds utilizing the essential range of the symbol and characterize the conditions under which $m(D_φ)$ coincides with the essential infimum of $|φ|$. In the general setting, for unimodular $\vp$, we obtain exact formulas and two sided estimates for $m(D_φ)$ by analyzing the norms of associated Toeplitz and Hankel operators restricted to the model space. Finally, we provide several concrete examples to illustrate our results. |
| title | On the minimum modulus of dual truncated Toeplitz operators |
| topic | Functional Analysis Spectral Theory 47B35, 47A30, and 47B32 |
| url | https://arxiv.org/abs/2602.15713 |