On the algebra generated by the Bergman projection and a shift operator. I (Q1402356)

Let \(G\subset\mathbb C\) be a domain with smooth boundary and let \(\alpha\) be a \(C^2\)-diffeomorphism on \(G\) satisfying the Carleman condition \(\alpha\circ\alpha = \text{id}_{\overline G}\). Let \(W\) stand for the shift operator \(W\varphi =\sqrt{|\det J_\alpha|} \varphi\circ\alpha\) acting on \(L_2(G, d\mu)\), where \(d\mu = dxdy\) is the usual Lebesgue measure. Denote the Bergman projection of \(G\) by \(K\). The most useful representation of \(K\) is \(K = I-S_GS_G^\ast+L,\) where \(L\) is a compact operator and \(S_G\) is the two-dimensional singular integral operator \[ (S_G\varphi)(z)=-\frac{1}{\pi}\int_G\frac{\varphi(\zeta)}{(\zeta-z)^2} d\mu(\zeta). \] The \(\mathbb C^\ast\)-algebra generated by a collection of operators \(\mathcal F\) and a finite number of operators \(A_1,\dots,A_m\) on a Hilbert space is denoted by \(\mathcal R(\mathcal F, A_1,\dots,A_m)\). Let \(\mathcal R_0 :=\mathcal R(C(\overline G)I; K,WKW)\). By means of local techniques, the authors describe the Calkin algebra of \(\mathcal R_0\) as the algebra of all continuous sections of a \(C^\ast\)-bundle \(\xi = (E,\rho,\overline G)\). Each local algebra \(\mathcal R_0(\zeta_0) =\rho^{-1}(\zeta_0)\) is generated by the local images of the orthogonal projections \(K\) and \(WKW\). The algebra generated by two orthogonal projections \(P_1\) and \(P_2\) is isomorphic to an algebra of \(2\times 2\) matrix-valued functions continuous on \(\Delta =\text{sp}(P_1 -P_2)^2\). Let \(H\in B(L_2(\mathbb R))\) denote the Hilbert transform \[ (H\varphi)(y)=\frac{1}{\pi i}\int_\mathbb R\frac{\varphi(t)}{t-y} dt. \] The Hardy spaces \(H_\pm^2(\mathbb R)\) are the images of the orthoprojections \(P_\pm =(I\pm H)/2\), respectively. The authors prove that there exists an operator \(A_{\zeta_0} = M_\lambda P_- + P_+I\), \(M_\lambda\in M_3(C(\dot\mathbb R))\), such that \(\lambda I - K + WKW\) is locally invertible at \(\zeta_0\in\partial G\) iff \(A_{\zeta_0}\) is invertible. Thus, the local spectrum of \(K- WKW\) at \(\zeta_0\in\partial G\) is computed by solving a Riemann boundary value problem. A symbol algebra of \(\mathcal R\) is determined and Fredholm conditions are given. It is proved that the \(C^\ast\)-algebra generated by the Bergman projection of the upper half-plane and the operator \((W\varphi)(z) =\varphi(-\overline z)\) is isomorphic and isometric to \(\mathbb C^2\times M_2(\mathbb C)\).