Journal article
The spectral constant for the quantum cross and asymptotically sharp bounds for annuli
Acta scientiarum mathematicarum (Szeged)
12 Jan 2026
Abstract
The quantum annulus of type r is the class of invertible operators with singular values in (1/r, r). Given an analytic function on the classical annulus of type r, we may evaluate it on operators in the quantum annulus via the holomorphic functional calculus. The spectral constant gives the maximum ratio betweeen the supremum over the norm of evalutions at operators in the quantum annulus to the supremum over classical evaluations. We show that the limit of the spectral constant as r goes to infinity is 2. Via the correspondence between annuli and hyperbolae, our study degenerates the problem to one on the quantum cross, pairs of contractions with product zero, where the spectral constant is exactly 2. As a consequence, we see the following elementary sharp Cartan extension theorem with bounds in the limit case: Let f,g:D -> D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f,g: \mathbb {D}\rightarrow \overline{\mathbb {D}}$$\end{document} be analytic functions such that f(0)=g(0).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(0)=g(0).$$\end{document} There is an h:D2 -> 2D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h: \mathbb {D}<^>2\rightarrow 2\overline{\mathbb {D}}$$\end{document} analytic such that h(z,0)=f(z),h(0,w)=g(w).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(z,0)=f(z), h(0,w)=g(w).$$\end{document} The essential technique is to rationally dilate Z to Z<^>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{Z}$$\end{document} which has U=(Z<^>+(Z<^>-1)& lowast;)/(r+1/r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U =(\hat{Z}+(\hat{Z}<^>{-1})<^>*)/(r+1/r)$$\end{document} unitary and estimate Uf(Z<^>)U & lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Uf(\hat{Z})U<^>*$$\end{document} directly.
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Details
- Title
- The spectral constant for the quantum cross and asymptotically sharp bounds for annuli
- Creators
- J. E. Pascoe - Drexel University, Mathematics
- Publication Details
- Acta scientiarum mathematicarum (Szeged)
- Publisher
- Springer Nature
- Number of pages
- 5
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:001659654900001
- Scopus ID
- 2-s2.0-105027326714
- Other Identifier
- 991022153554104721
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- Mathematics