The arithmetic–geometric mean and Bohr-type inequalities for sectorial matrices

Abstract

In this paper, we establish arithmetic–geometric mean inequalities for products of sectorial matrices under arbitrary unitarily invariant norms. For \(\alpha \)-sectorial operators A,B and a positive semidefinite matrix X, we bound the norm of AXB by an expression involving \((\operatorname {Re}A)^{2}\) and \((\operatorname {Re}B)^{2}\) with the multiplicative factor \(\sec ^{2}\alpha \). We develop trace versions, extend the results to positive linear maps, and establish matrix Bohr inequalities for sectorial matrices. These results encompass the classical case of positive semidefinite matrices when \(\alpha =0\). We provide explicit equality examples demonstrating that our estimates are sharp.