P-Essential Normality Of Quasi-Homogeneous Hilbert Modules

In the present paper, we focus on essential normality of Hilbert modules over the complex unit ball Bd. In the beginning, when searching for commutative solu-tions for homogeneous equations, W. Arveson raised the conjecture that whether or not each homogeneous submodule of the symmetric Fock space Hd2 is p-essentially normal for p>d. From then on, Arveson’s conjecture and invariants on varies of Hilbert modules became the principle problem in geometry and analysis on Hilbert modules. Essential normality of Hilbert modules is in deep link to other branches of mathematics, such as algebraic geometry, index theory, and C*-extension theory, etc.We make use of quasi-homogeneity and get rid of the dependence on measure, to prove p(>d)-essential normality of principal quasi-homogeneous submodules, and as a consequence prove the p(>3)-essential normality of every quasi-homogeneous submodule in the case d= 3. Based on these results, we prove that the K-homology invariants for quasi-homogeneous quotient modules are nontrivial when d=3, which is connected to index theory.On the Bergman module La2(Ed) or the Hardy module H2((?)Bd), denote by Tφ the Toeplitz operator of analytic symbol φ. We prove by trace estimate that, bounded operators of the form A=Σk TφkTφk* are p(> 2d)-essentially commutable with each Tzi(i=1,…,d), and the p(> 2d)-norm of [A,Tzi]is controlled by the operator norm of A. This conclusion is extended to (2d, ∞)-essential commutativity and (2d, oo)-norm, and the Dixmier trace of |[A, TZi]|2d is estimated.If an operator is essentially equivalent to a bounded operator of form ΣkTφkTφk*, then it is called approximately representable. We prove that p-approximate repre-sentability of a homogeneous Bergman submodule is intrinsically equivalent to its p-essential normality, and therefore find a new tool for the study of essential nor-mality of homogeneous submodules.