The numerical radius of fractional powers of matrices

Using integral representations of the fractional power of matrices, and the geometric intuition of sectorial matrices, we show that for any accretive-dissipative matrix $A$ and any $t\in (0,1)$, the matrix $A^t$ is accretive-dissipative, and that $$\omega (A^t)\ge {\omega }^t(A),$$ where $\omega (\cdot )$ is the numerical radius. This inequality complements the well-known power inequality $\omega (A^k)\le {\omega }^k(A)$, valid for any square matrix and positive integer power $k$. As an application, we prove that if $A$ is accretive, then the above fractional inequality holds if $0<t<\frac{1}{2}$. Other consequences will be given too.