Numerical ranges and complex symmetric operators in semi-inner-product spaces

被引：0
作者：
Il Ju An
Jaeseong Heo
机构：
[1] Kyung Hee University,Department of Applied Mathematics
[2] Hanyang University,Department of Mathematics, Research Institute for Natural Sciences
来源：
Journal of Inequalities and Applications | / 2022卷
关键词：
Semi-inner-product space; Numerical range; Conjugations; Complex symmetric operators; Generalized adjoint; 46C50; 47A05; 47A12;
D O I：
暂无
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摘要：
We introduce the numerical range of a bounded linear operator on a semi-inner-product space. We compute the numerical ranges of some operators on ℓ2p(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell _{2}^{p}(\mathbb{C})$\end{document}(1≤p<∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(1\le p < \infty )$\end{document} and show that the numerical range of the backward shift on an infinite-dimensional space ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell ^{p}$\end{document} is the open unit disc. We define a conjugation and a complex symmetric operator on a semi-inner-product space and discuss complex symmetry in the dual space. We prove some properties of a generalized adjoint of a complex symmetric operator. We also show that the numerical range of the complex conjugation on ℓnp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell _{n}^{p}$\end{document}(n≥2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(n \ge 2)$\end{document} is the closed unit disc. Finally, we discuss the sequentially essential numerical ranges of operators on a semi-inner-product space.
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共 25 条
[1]  
Chō M.(2019)Complex symmetric operators and isotropic vectors on Banach spaces J. Math. Anal. Appl. 479 752-764
[2]  
Hur I.(2018)On conjugations for Banach spaces Sci. Math. Jpn. 81 37-45
[3]  
Lee J.E.(2006)Quantum error correcting codes from the compression formalism Rep. Math. Phys. 58 77-91
[4]  
Chō M.(1972)On the essential numerical range, the essential spectrum, and a problem of Halmos Acta Sci. Math. 33 179-192
[5]  
Tanahashi K.(2014)Mathematical and physical aspects of complex symmetric operators J. Phys. A 47 436-446
[6]  
Choi M.-D.(1967)Classes of semi-inner product spaces Trans. Am. Math. Soc. 129 64-65
[7]  
Kribs D.W.(1965)On the representation of a unitary operator in the form of a product of two involutions Usp. Mat. Nauk 20 2990-2997
[8]  
Życzkowski K.(2021)Numerical ranges of conjugations and antilinear operators Linear Multilinear Algebra 69 363-366
[9]  
Fillmore P.A.(1971)A note on some operator theory in certain semi-inner-product spaces Proc. Am. Math. Soc. 30 29-43
[10]  
Stampfli J.G.(1961)Semi-inner-product spaces Trans. Am. Math. Soc. 100 83-93
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