Improved inequalities for numerical radius via cartesian decomposition

We derive various lower bounds for the numerical radius $w(A)$ of a bounded linear operator $A$ defined on a complex Hilbert space, which improve the existing inequality $w^2(A)\geq \frac{1}{4}\|A^*A+AA^*\|$. In particular, for $r\geq 1$, we show that
$$\tfrac{1}{4}\|A^*A+AA^*\|\leq\tfrac{1}{2}(\tfrac{1}{2}\|\operatorname{Re}(A)+\operatorname{Im}(A)\|^{2r}+\tfrac{1}{2}\|\operatorname{Re}(A)-\operatorname{Im}(A)\|^{2r})^{1/r} \leq w^{2}(A),$$
where $\operatorname{Re}(A)$ and $\operatorname{Im}(A)$ are the real and imaginary parts of $A$, respectively. Furthermore, we obtain upper bounds for $w^2(A)$ refining the well-known upper estimate $w^2(A)\leq \frac{1}{2}(w(A^2)+\|A\|^2)$. Criteria for $w(A)=\frac12\|A\|$ and for $w(A)=\frac{1}{2}\sqrt{\|A^*A+AA^*\|}$ are also given.