Spectral radius of semi-Hilbertian space operators and its applications
By Kais Feki. Annals of Functional Analysis, 2019
112 citations recorded by the source corpus.
In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}$$\end{document}, which are bounded with respect to the seminorm induced by a positive operator A on H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}$$\end{document}. Mainly, we show that rA(T)≤ωA(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_A(T)\le \omega _A(T)$$\end{document} for every A-bounded operator T, where rA(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_A(T)$$\end{document} and ωA(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _A(T)$$\end{document} denote respectively the A-spectral radius and the A-numerical radius of T. This allows to establish that rA(T)=ωA(T)=‖T‖A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_A(T)=\omega _A(T)=\Vert T\Vert _A$$\end{document} for every A-normaloid operator T, where ‖T‖A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert T\Vert _A$$\end{document} is denoted to be the A-operator seminorm of T. Moreover, some characterizations of A-normaloid and A-spectraloid operators are given.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/54bf7d7b7263056177ca16735503b57875d82265
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Citations in the corpus
- A-Numerical Radius Orthogonality and Parallelism of Semi-Hilbertian Space Operators and Their Applications (2020, background): “Recall from [14] that an operator T ∈ BA1/2(H) is said to be A-normaloid if rA(T ) = ‖T ‖A, where rA(T ) = lim n→+∞ ‖T n‖ 1 n A.”
- A note on the A-numerical radius of operators in semi-Hilbert spaces (2020, background): “Moreover, the following inclusions become BA(H) ⊆ BA1/2(H) ⊆ B(H) hold and are in general proper (see [7]).”