Quotient modules for some Hilbert modules over the bidisk (Q961045)

Let \((T_1,\dots,T_d)\) be a \(d\)-tuple of bounded linear operators on a Hilbert space \(\mathcal H\). The Hilbert space \(\mathcal H\) has a Hilbert module structure over the polynomial ring \({\mathbb C}[z_1,\dots,z_d]\) defined by the formula \[ p\cdot\xi=p(T_1,\dots,T_d)\xi, \quad p\in{\mathbb C}[z_1,\dots,z_d], \;\xi\in{\mathcal H}. \] If all cross-commutators \(T^\ast_kT_j-T_jT^\ast_k\), \(1\leqslant j,k\leqslant d\), are compact, then the Hilbert module is called essentially normal. Let \(H=H^{(\alpha,\beta)}\) be the Hilbert module over the bidisk with the reproducing kernel \[ K_\lambda(z)={1\over (1-z_1\overline{\lambda}_1)^\alpha (1-z_2\overline{\lambda}_2)^\beta}, \] where \(\alpha, \beta>0\) and \(\lambda=(\lambda_1,\lambda_2)\), \(z=(z_1,z_2)\in\mathbb D^2\). Let \(M=M^{(\alpha,\beta)}\) be the submodule generated by the quasi-homogeneous polynomials \(z_1^n-z_2^m\) (\(n,m\) any positive integers) and let \(H^{(\alpha,\beta)}/M^{(\alpha,\beta)}\) be its associated quotient module which is denoted by \(N=N^{(\alpha,\beta)}\). Let \(M_{z_1}\), \(M_{z_2}\) be the multiplication operators on \(H\). Then the module action on the quotient module \(N\) is endowed by the tuple \((S_{z_1},S_{z_2})\), where \(S_{z_i}\) is the compression of \(M_{z_i}\) to \(H\ominus M\), \(i=1,2\). The main result of the paper is the following: Theorem. The quotient module \(N\) is essentially normal, i.e., the operators \(S^\ast_kS_j-S_jS^\ast_k\), \(j,k=1,2\), are compact. The author also describes the essential spectrum of the tuple \((S_{z_1},S_{z_2})\) and shows that the Toeplitz algebra which is a \(C^\ast\)-algebra generated by this tuple is irreducible.