Product equivalence of quasihomogeneous Toeplitz operators on the harmonic Bergman space (Q2866764)

Let \(dA\) denote the Lebesgue area measure on the unit disk \(D\), normalized so that the measure of \(D\) equals 1, and let \(L^2(D,dA)\) be the Hilbert space of Lebesgue square integrable functions on \(D\). The harmonic Bergman space \(L^2_h\) is the closed subspace of \(L^2(D,dA)\) consisting of the harmonic functions on \(D\). The orthogonal projection from \(L^2(D,dA)\) onto \(L^2_h\) is denoted by \(Q\). Given \(z \in D\), let \(K_z(w) = 1/(1-w\bar{z})^2\) be the well-known reproducing kernel for the analytic Bergman space \(L^2_a\) consisting of all \(L^2\)-analytic functions on \(D\). The well-known Bergman projection \(P\) is then the integral operator \(Pf(z)= \int_D f(w)\overline{K_z(w)}\, dA(w)\) for \(f \in L^2(D,dA)\). Thus, \(Q\) can be represented by \(Qf = Pf + \overline{P\bar{f}} - Pf(0)\). For \(u \in L^1(D,dA)\), the Toeplitz operator \(T_u\) with symbol \(u\) is the operator on \(L^2_h\) defined by \(T_u f = Q(uf)\) for \(f \in L^2_h\). This operator is always densely defined on the polynomials and not bounded in general. The authors are interested in the case where it is bounded in the \(L^2_h\) norm, and \(u\) is a \(T\)-function. Then \(T_u\) has the continuous extension. A function \(f\) is said to be quasihomogeneous of degree \(k \in \mathbb{Z}\) if \(f(re^{i\theta})= e^{ik\theta}\varphi(r)\), where \(\varphi\) is a radial function. Let \(f_1\) and \(f_2\) be two quasihomogeneous \(T\)-functions on \(D\). In this case, the authors prove that, if there exists a \(T\)-function \(f\) such that \(T_{f_1}T_{f_2}=T_f\), then \(T_{f_2}T_{f_1}=T_f\) and \(T_{f_1}T_{f_2}=T_{f_2}T_{f_1}\).