Some A-spectral radius inequalities for A-bounded Hilbert space operators

Let 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} denote the A-spectral radius of an operator T which is bounded with respect to the seminorm induced by a positive operator A 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}. In this paper, we aim to establish several A-spectral radius inequalities for products, sums and commutators of A-bounded operators. Some applications of our results are provided. Moreover, we give an affirmative answer to the question recently posed by Baklouti and Namouri (Banach J Math Anal 16:12, https://doi.org/10.1007/s43037-021-00167-1, 2022) regarding the connection between the notions of A-spectral radius and A-spectrum for A-bounded operators.