Trace formulas and \(p\)-essentially normal properties of quotient modules on the bidisk (Q2891157)

Let \(H=H_\nu\) be the functional Hilbert space on the unit disc \(\mathbb D\) with the reproducing kernel \(K_w(z)=\frac{1}{(1-z\bar{w})^\nu}\) for \(\nu\geq 1\). It is the Hardy space \(H^2(\mathbb T)\;(\nu=1)\) or the weighted Bergman space \(L^2_a(\mathbb D,d\mu_{\nu-2})\) \((\nu>1)\), where \(d\mu_\alpha=c_\alpha(1-| z|^2)^\alpha dm(z)\) is the normalized measure on \(\mathbb D\). The space \(H\otimes H\) is then the Hilbert space \(H^2(\mathbb T^2)\) or \(L^2_a(\mathbb D^2, d\mu_{\nu-2}\times d\mu_{\nu-2})\) on the bidisc \(\mathbb D^2\). Let \(M_f\) be the multiplication operator on \(H\otimes H\) for \(f\in H^\infty(\mathbb D)\). For an invariant subspace \(\mathcal M\) of the multiplication operators \(M_z, M_w\) on \(H\otimes H\) denote by \(S_f=P_{\mathcal M^\bot}M_f P_{\mathcal M^\bot}\) and \(H_f=P_{\mathcal M}M_fP_{\mathcal M^\bot}\), respectively, the compression of \(M_f\) to the quotient module \(\mathcal M^\bot\) and the Hankel type operator. The authors study properties of commutators \([S^\ast_f,S_f]\) and, among others, prove the following:NEWLINENEWLINE \noindent Theorem 1. The commutators \([S^\ast_z,S_z]\), \([S^\ast_w,S_w]\), \([S^\ast_z,S_w]\) are of trace class. Thus the quotient module \(\mathcal M^\bot\) is in \(\mathcal L^1\) (trace class operators).NEWLINENEWLINE They obtain the following trace formula:NEWLINENEWLINE \noindent Theorem 2. Let \(F(z,w)\) be a polynomial in \((z,w)\) and \(f(z)=F(z,z)\) be its restriction to the diagonal. Then NEWLINE\[NEWLINE{\text{Tr}}\,[S^\ast_F,S_F]=(N+1)\int_{\mathbb D}| f'(z)|\,dm(z).NEWLINE\]NEWLINE Similar trace formulas for Hankel type operators are also obtained.