Further improvements of generalized numerical radius inequalities for Hilbert space operators

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 $$\gamma (T)=\sqrt{{(\mathrm{\parallel }{\mathrm{\Re }}_A(T){\mathrm{\parallel }}_A^2-\mathrm{\parallel }{\mathrm{\Im }}_A(T){\mathrm{\parallel }}_A^2)}^2+4\mathrm{\parallel }{\mathrm{\Re }}_A(T){\mathrm{\Im }}_A(T){\mathrm{\parallel }}_A^2}\mathrm{.}$$ Here ${\omega }_A(X)$ and $\mathrm{\parallel }X{\mathrm{\parallel }}_A$ denote respectively the $A$-numerical radius and the $A$-seminorm of an operator $X$. Also, ${\mathrm{\Re }}_A(T):=\frac{T+T^{{\mathrm{♯}}_A}}{2}$ and ${\mathrm{\Im }}_A(T):=\frac{T-T^{{\mathrm{♯}}_A}}{2i}$, where $T^{{\mathrm{♯}}_A}$ is a distinguished $A$-adjoint operator of $T$. Further, some new refinements of the triangle inequality related to $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_A$ are established.