We develop 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 \\\\begin{eqnarray*}\\\\frac{1}{4}\\\\|A^*A+AA^*\\\\| \\\\leq\\\\frac{1}{2} \\\\left( \\\\frac{1}{2}\\\\|\\\\Re(A)+\\\\Im(A)\\\\|^{2r}+\\\\frac{1}{2}\\\\|\\\\Re(A)-\\\\Im(A)\\\\|^{2r}\\\\right)^{\\\\frac{1}{r}} \\\\leq w^{2}(A),\\\\end{eqnarray*} where $\\\\Re(A)$ and $\\\\Im(A)$ are the real and imaginary parts of $A$, respectively. Furthermore, we obtain upper bounds for $w^2(A)$ refining the well-known upper bound $w^2(A)\\\\leq \\\\frac{1}{2} \\\\left(w(A^2)+\\\\|A\\\\|^2\\\\right)$. Separate complete characterizations for $w(A)=\\\\frac{\\\\|A\\\\|}{2}$ and $w(A)=\\\\frac{1}{2}\\\\sqrt{\\\\|A^*A+AA^*\\\\|}$ are also given.

Keywords