Numerical Range of the Linear Combination of Volterra Operator and Its Adjoint
Received:April 21, 2022  Revised:June 27, 2022
Key Words: Volterra operator   the adjoint operator   numerical range
Fund Project:Supported by the Tianjin Normal University Research Innovation Project for Postgraduate Students (Grant No.2022KYCX110Y).
 Author Name Affiliation Jinxian WANG School of Mathematical Science, Tianjin Normal University, Tianjin 300387, P. R. China Yuxia LIANG School of Mathematical Science, Tianjin Normal University, Tianjin 300387, P. R. China
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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.