The numerical radius of fractional powers of matrices

Abstract

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 \[ \omega(A^t)\geq \omega^t(A) , \] where \(\omega(\cdot)\) is the numerical radius. This inequality complements the well-known power inequality $\omega(A^k)\leq \omega^k(A)$, 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.