Curvature inequalities for operators in the Cowen-Douglas class and localization of the Wallach set

Abstract

For any bounded domain in C-m, let B-1() denote the Cowen-Douglas class of commuting m-tuples of bounded linear operators. For an m-tuple T in the Cowen-Douglas class B-1(), let N-T (w) denote the restriction of T to the subspace i,j=1m ker(Ti-wiI)(Tj-wjI). This commuting m-tuple N-T (w) of m + 1 dimensional operators induces a homomorphism NT(w) of the polynomial ring Pz(1), , z(m)], namely, NT(w) (p) = p(N-T (w)), p Pz(1), , z(m)]. We study the contractivity and complete contractivity of the homomorphism NT(w). Starting from the homomorphism NT(w), we construct a natural class of homomorphisms N()(w), > 0, and relate the properties of N()(w) to those of NT(w). Explicit examples arising from the multiplication operators on the Bergman space of are investigated in detail. Finally, it is shown that contractive properties of NT(w) are equivalent to an inequality for the curvature of the Cowen-Douglas bundle E-T. However, we construct examples to show that the contractivity of the homomorphism (T) does not follow, even if NT(w) is contractive for all w in .