Operator Arithmetic-Geometric-Harmonic mean inequality on Krein spaces

Mahdi Dehghani, Seyed Mohammad sadegh Modarres Mosadegh


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:
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


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

Full Text: PDF


  • There are currently no refbacks.

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.