Norm inequalities in operator ideals

In this paper we introduce a new technique for proving norm inequalities in operator ideals with a unitarily invariant norm. Among the well-known inequalities which can be proved with this technique are the Löwner–Heinz inequality, inequalities relating various operator means and the Corach–Porta–Recht inequality. We prove two general inequalities and from them we derive several inequalities by specialization, many of them new. We also show how some inequalities, known to be valid for matrices or bounded operators, can be extended with this technique to normed ideals in C ∗-algebras, in particular to the noncommutative Lp-spaces of a semi-finite von Neumann algebra.