Essentially Reductive Hilbert Modules II

Many Hilbert modules over the polynomial ring in m variables are essentially reductive, that is, have commutators which are compact. Arveson has raised the question of whether the closure of homogeneous ideals inherit this property and provided motivation to seek an affirmative answer. Positive results have been obtained by Arveson, Guo, Wang and the author. More recently, Guo and Wang extended the result to quasi-homogeneous ideals in two variables. Building on their techniques, in this note the author extends this result to Hilbert modules over certain Reinhardt domains such as ellipsoids in two variables and analyzes extending the result to the closure of quasi-homogeneous ideals in m variables when the zero variety has dimension one. 0 Introduction In his study [1] of the m-shift Hilbert module H m over C[z ], the polynomial algebra in m variables, Arveson formulated a provocative conjecture which has attracted the attention of several researchers. He had established that the commutators of all operators and their adjoints defined on H m by module multiplication by polynomials in C[z ] belonged to the Schatten pclass Lp for all p > m. He then conjectured that the same commutator condition held for all submodules in H m obtained as the closure [I] in H 2 m of a homogeneous ideal I and established the result in [2] in case I is generated by monomials. Using somewhat different methods, the author extended this latter result in [10] to a larger family of commuting weighted shifts in The same result was proved earlier for the quotient defined by every homogeneous submodule in the Hardy space H (D) for the bidisk D by Curto, Muhly and Yan [9]. Such Hilbert modules were defined to be essentially reductive in [12] and, later, essentially normal in [2]. This research was prompted by discussions with K. Guo and K. Wang during a visit to Fudan University in July, 2005 and with Arveson and other researchers during a DST-NSF funded visits to Bangalore in December, 2003 and 2005. Mathematics Subject Classification: 32T15, 46L80, 46M20, 47L15.