Numerical Ranges of Radial Toeplitz Operators on Bergman Space

A Toeplitz operator T_fT_phi with symbol fphi in L^￥(mathbbD)L^{infty}({mathbb{D}}) on the Bergman space A²(mathbbD)A^{2}({mathbb{D}}), where mathbbDmathbb{D} denotes the open unit disc, is radial if f(z) = f(|z|)phi(z) = phi(|z|) a.e. on mathbbDmathbb{D}. In this paper, we consider the numerical ranges of such operators. It is shown that all finite line segments, convex hulls of analytic images of mathbbDmathbb{D} and closed convex polygonal regions in the plane are the numerical ranges of radial Toeplitz operators. On the other hand, Toeplitz operators T_fT_phi with fphi harmonic on mathbbDmathbb{D} and continuous on [`(mathbbD)]{overline{mathbb{D}}} and radial Toeplitz operators are convexoid, but certain compact quasinilpotent Toeplitz operators are not.