Pluriharmonic symbols of essentially commuting Toeplitz operators
Fix a positive integer \(n\) and consider the open unit ball \(B\) in the complex Euclidean space \(\mathbb{C}^n\). Let \(A^2\) denote the Bergman \(L^2\)-space on \(B\). For a bounded function \(\varphi\) in \(B\) let \(T_\varphi\) denote the Toeplitz operator on the Bergman space \(A^2\). Suppose \(u\) and \(v\) are two bounded pluriharmonic functions on \(B\) with \(u= f+\overline g\) and \(v= h+\overline k\), where \(f\), \(g\), \(h\), and \(k\) are holomorphic in \(B\). The main result of the paper states that the following conditions are equivalent. (a) \(T_u\) and \(T_v\) are essentially commuting on \(A^2\), that is, the commutant \(T_uT_v- T_vT_u\) is compact. (b) \(\widetilde\Delta(f\overline k- h\overline g)(z)\to 0\) as \(z\to\partial B\). (c) The integrals \[ \int_B|(f\circ \varphi_a- f(a))(\overline k\circ \varphi_a-\overline k(a))- (h\circ\varphi_a- h(a))(\overline g\circ\varphi_a-\overline g(a))| dV \] approach zero as \(a\to\partial B\), where \(\varphi_a\) is the involutive automorphism of \(B\) that interchanges \(a\) and \(0\). The paper also obtains an equivalent condition in terms of the maximal ideal space of \(H^\infty(B)\).
scientific article; zbMATH DE number 1191162