Refinements of Inequalities Related to Landau–Grüss Inequalities for Elementary Operators Acting on Ideals Associated to p-Modified Unitarily Invariant Norms

In this paper we find the explicit form for the equalizing term in the inequality related to Landau–Gruss inequality for elementary operators, which enabled us to refine inequalities related to operator Landau–Gruss norm inequalities for p-modifications of initarily invariant norms. Specially, for Schatten p-norms with \(p\ge 2\) those inequalities have additionally simplified form, which enables, among others, to analyze equality cases in considered operator inequalities. A broader family of mixed norms inequalities related to the operator Landau–Gruss inequality is also presented.