Mohammad Sababheh, Hamid Reza Moradi, On the real parts of a matrix and its Aluthge transform, Vol. 2025 (2025), No. 23, pp. 1-16

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DOI: 10.23952/jnfa.2025.23

Received December 6, 2024; Accepted June 20, 2025; Published August 1, 2025

Abstract. Our goal in this paper is to discuss possible bounds and relations for the real part of a complex matrix. In particular, we present several relations between the real parts of the matrix and its Aluthge transform. We show that if $T$ is an $n\times n$ complex matrix, then
$\|RT\|\le \frac{1}{2}\max \{\| |T|+R\widetilde{T}\|,\| |T|-R\widetilde{T}\|\}\le \frac{1}{2}\| |T^*|+|T|\|,$
where $\|\cdot\|, |\cdot|, \Re(\cdot), \widetilde{T}$ denote the usual operator norm, the absolute value, the real part, and the Aluthge transform, respectively. We also present numerical radius bounds, a spectral radius identity, and an arithmetic-geometric mean inequality for the real part and the numerical radius. In particular, we show that if $A,B$ are $n\times n$ complex matrices, then
$\|R(AB)\|=\frac{1}{2}\rho \Big(\big[\begin{matrix} BA & \frac{1}{t^2} |B^*|^2\\ t^2|A|^2 & (BA)^*\\\end{matrix} \big] \Big),$
where $\rho(\cdot)$ is the spectral radius, and $t>0$ is arbitrary.

How to Cite this Article:
M. Sababheh, H.R. Moradi, On the real parts of a matrix and its Aluthge transform, J. Nonlinear Funct. Anal. 2025 (2025) 23.