The numerical range and normality of Toeplitz operators (Q2731574)

Let \(B\) be the unit ball of \(\mathbf C^n\), \(A^2\) the Bergman space of all holomorphic functions in \(L^2(B)\), and for \(u\) a bounded measurable function on \(B\) let \(T_u:f\mapsto P(uf)\), where \(P\) stands for the orthogonal projection of \(L^2(B)\) onto \(A^2\), be the Toeplitz operator on \(A^2\) with symbol \(u\). Recall that a function \(u\) on \(B\) is called \(M\)-harmonic if it is annihilated by the invariant Laplacian \(\widetilde \Delta\) on \(B\) (i.e. \(\Delta(u\circ\varphi)(0)=0\) for all holomorphic automorphisms \(\varphi\) of \(B\)), and pluriharmonic if its restriction to any complex line is a harmonic function of one complex variable. The main results of the paper [generalizing those of \textit{J. K. Thukral} for the unit disc, J. Oper. Theory 34, No. 2, 213-216 (1995; Zbl 0851.47020)]) are the following: NEWLINENEWLINENEWLINEIf \(u\) is a bounded \(M\)-harmonic function on \(B\) and the numerical range \(W(T_u)\) of \(T_u\) is not open in \(\mathbf C\), then \(T_u\) is a normal operator (Theorem 4); NEWLINENEWLINENEWLINEif \(u\in H^\infty(B)\), then \(W(T_u)\) is the convex hull of \(u(B)\) (Corollary 6); NEWLINENEWLINENEWLINEif \(u\) is a nonconstant, real-valued, bounded \(M\)-harmonic function on \(B\), then \(W(T_u)\) is the open interval \((\inf u,\sup u)\) (Theorem 9); NEWLINENEWLINENEWLINEand if \(u\) is a nonconstant, bounded pluriharmonic functions on \(B\), then \(T_u\) is normal if and only if \(W(T_u)\) is an open line segment (Theorem 11). NEWLINENEWLINENEWLINEFinally, similar assertions are also obtained for \(A^2\) replaced by the pluriharmonic Bergman space \(b^2\) on \(B\).