Numerical Range of the Linear Combination of Volterra Operator and Its Adjoint

Abstract:

      The classical Volterra operator $V$ and its adjoint operator $V^*$ play important roles in the complex space $L^2[0,1]$. As to the properties of linear combination of $V$ and $V^*$, we present the equivalent condition ensuring $z_1V+z_2V^*(z_1,z_2\in\mathbb{C})$ satisfies the accretive property. Then an accurate representation of the numerical range of $(u+iv)I+mV+nV^*\,(u,v,m,n\in\mathbb{R},m+n\geq0)$ is described in this paper.