J-selfadjoint matrix means and their indefinite inequalities

Abstract

AbstractLet J be a non trivial involutive Hermitian matrix. Consider $${\mathbb {C}}^n$$ C n equipped with the indefinite inner product induced by J, $$[x,y]=y^*J x$$ [ x , y ] = y ∗ J x for all $$x,y\in {{\mathbb {C}}}^n,$$ x , y ∈ C n , which endows the matrix algebra $${\mathbb {C}}^{n\times n}$$ C n × n with a partial order relation $$\le ^J$$ ≤ J between J-selfadjoint matrices. Inde-finite inequalities are given in this setup, involving the J-selfadjoint $$\alpha $$ α -weighted geometric matrix mean. In particular, an indefinite version of Ando–Hiai inequality is proved to be equivalent to Furuta inequality of indefinite type.