Operator Arithmetic-Geometric-Harmonic mean inequality on Krein spaces

Abstract

Let $J$ be a selfadjoint involution, i.e. $J=J^*=J^{-1}$, on a
Hilbert space $\mathscr{H}$. We prove an operator
arithmetic-harmonic mean inequality for invertible $J$-positive
operators $A$ and $B$ on the Krein space $(\mathscr{H},J)$, by
using some block matrix techniques of indefinite type, as follows:
\begin{eqnarray*}
A!_{\lambda}B\leq^{J}A\nabla_{\lambda}B\qquad(\lambda\in[0,1]),
\end{eqnarray*}
where $A\nabla_{\lambda}B=\lambda A+(1-\lambda)B$ and
$A!_{\lambda}B=(\lambda A^{-1}+(1-\lambda)B^{-1})^{-1}$ are
arithmetic and harmonic mean of $A$ and $B$, respectively. We also
give an example which shows that the operator
arithmetic-geometric-harmonic mean inequality for two invertible
$J$-selfadjoint operators on Krein spaces is not valid, in
general.