### 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.

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.

#### Keywords

block matrix, $J$-selfadjoint operator, $J$-positive operator, operator arithmetic-geometric-harmonic mean inequality, operator mean

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