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Product equivalence of quasihomogeneous Toeplitz operators on the harmonic Bergman space. (English) Zbl 1310.47041
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 $$T_{f_1}T_{f_2}=T_{f_2}T_{f_1}$$.

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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