Advances in Operator Theory and Inequalities

Abstract:
This paper reviews some of the results obtained by the authors in their recent work on matrix inequalities involving sectorial matrices and a matrix monotone function defined by the harmonic mean integral. The study focuses on the important concept of matrix inequalities that relate to the Schur complement of sectorial matrices with respect to a matrix monotone function.
if $A$ and $B$ are two matrices such that $ W(A) , W(B) \subseteq \mathcal{S}_{\alpha} $ for some $\alpha \in [0,\frac{\pi}{2} )$ and $\sigma_{f} $ is a matrix mean, then
\begin{align}
\Re(S(A) \sigma_{f} S(B)) \leq (\sec \alpha)^{4} \Re\textit{S}(A\sigma_{f} B).
\end{align}
Moreover, it is obtained
\begin{align}
\Re (f(S(A))) \leq (\sec \alpha)^{4} \Re S(f(A)),
\end{align}
where $f$ is a non-negative matrix monotone function with $f(1)=1$ and $A$ is a matrix such that $ W(A) \subseteq \mathcal{S}_{\alpha} $ for some $\alpha \in [0,\frac{\pi}{2} )$.