Trace estimation of commutators of multiplication operators on function spaces (Q384330)

Let \(L^2_a(B_d)\) and \(H^2(B_d)\) be the Bergman space and Hardy space on the \(d\)-dimensional complex unit ball \(B_d\) and \(\phi_k\) be the multiplier for any integer \(k\). These functional spaces as Hilbert modules have a natural \(C[z_1,z_2,\dots,z_d]\) module structure. In [Acta Math. 181, No. 2, 159--228 (1998; Zbl 0952.46035)], \textit{W. Arveson} conjectured that the graded sumodules of the Drury-Arveson module on \(B_d\) are \(P\)-essentially normal for any \(p>d\). \textit{K.-Y. Guo} and \textit{K. Wang} [Math. Ann. 340, No. 4, 907--934 (2008; Zbl 1148.47005)] proved the case \(d=2,3\) of this conjecture. Moreover, the \(p\)-essential normality of the submodule \(M\) can be deduced from the commutator of \(P_M\) and the multiplier operator \(M_{z_i}\) belongs to \(L_{2p}\) (see Arveson [loc.\,cit.]). In spite of this, the authors prove that, for any operator \(A=\sum_{k\geq 1} T_{\phi_k}T^*_{\phi_k}\), the commutator \([A,T_{z_i}]\) belongs to the Schatten class \(L_{2p}\), \(p>d\), and that NEWLINE\[NEWLINE\|[A,T_{z_i}]\|_{2p}\leq C\|A\|,NEWLINE\]NEWLINE where \(C\) only depends on \(p\) and \(d\).