On some eigenvalue inequalities for Schatten–von Neumann operators

Michael Gil' recently obtained some bounds for eigenvalues in J. Funct. Anal. 267 (2014) 3500–3506] and Commun. Contemp. Math. 18 (2016) 1550022], which improve some classical results related to this aspect. We revisit these results by providing genuinely different arguments (e.g., using Aluthge transform, majorization). New results are derived along our discussions.