MMN-3774

# Further improvements of generalized numerical radius inequalities for Semi-Hilbertian space operators

**Kais Feki**, University of Monastir, Faculty of Economic Sciences and Management of Mahdia, Mahdia, Tunisia Laboratory Physics-Mathematics and Applications (LR/13/ES-22), University of Sfax, Faculty of Sciences of Sfax, Sfax, Tunisia,`kais.feki@hotmail.com ; kais.feki@fsegma.u-monastir.tn`

## Abstract

Several new improvements of the $A$-numerical radius inequalities for operators acting on a semi-Hilbert space, i.e., a space generated
by a positive operator $A$, are proved. In particular, among other inequalities, we show that
\begin{align*}
\frac{1}{4}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A
\leq\frac{1}{4}\Big(2\omega_A^2(T)+\gamma(T)\Big)
\leq \omega_A^2(T),
\end{align*}
where
$$\gamma(T)=\sqrt{\left(\|\Re_A(T)\|_A^2-\|\Im_A(T)\|_A^2\right)^2+4\|\Re_A(T)\Im_A(T)\|_A^2}.$$
Here $\omega_A(X)$ and $\|X\|_A$ denote respectively the $A$-numerical radius and the $A$-seminorm of an operator $X$. Also, $\Re_A(T):=\frac{T+T^{\sharp_A}}{2}$ and $\Im_A(T):=\frac{T-T^{\sharp_A}}{2i}$, where $T^{\sharp_A}$ is a distinguished $A$-adjoint operator of $T$. Further, some new refinements of the triangle inequality related to $\|\cdot\|_A$ are established.

Vol. 23 (2022), No. 2, pp. 651-665

DOI: 10.18514/MMN.2022.3774