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.