Essential normality of Bergman modules over intersections of complex ellipsoids

Abstract

A commuting tuple (T1, . . . , Tm) of operators, also called a multioperator, on a Hilbert space H is called essentially normal if all of the commutators [Tj , T ∗ k ], j, k = 1, . . . ,m are compact. Alternatively, essential normality can be attributed to the Hilbert C[z1, . . . , zm]module generated by (T1, . . . , Tm), namely, H equipped with the module action P (z1, . . . , zm)· f , P ∈ C[z1, . . . , zm], f ∈ H given by P (T1, . . . , Tm)f . Brown, Douglas and Fillmore [12, 13, 19] classified essentially normal multioperators up to unitary equivalence. The complete classifier here is the odd K-homology functor K1 from the category of compact metrizable spaces to the category of abelian groups. More precisely, for any compact subspace X ⊆ Cm, the abelian group K1(X) classifies essentially normal multioperators with essential Taylor spectrum X up to unitary equivalence; the elements of K1(X) are equivalence classes of C*monomorphisms from C(X) to the algebra of bounded operators on H modulo the ideal of