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.
Funder
Fundação para a Ciência e a Tecnologia
Universidade de Aveiro
Publisher
Springer Science and Business Media LLC
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