Contractivity, complete contractivity and curvature inequalities

Abstract

For any bounded domain $\Omega$ in $\mathbb C^m,$ let ${\mathrm B}_1(\Omega)$ denote the Cowen-Douglas class of commuting $m$-tuples of bounded linear operators. For an $m$-tuple $\boldsymbol T$ in the Cowen-Douglas class ${\mathrm B}_1(\Omega),$ let $N_{\boldsymbol T}(w)$ denote the restriction of $\boldsymbol T$ to the subspace ${\cap_{i,j=1}^m\ker(T_i-w_iI)(T_j-w_jI)}.$ This commuting $m$-tuple $N_{\boldsymbol T}(w)$ of $m+1$ dimensional operators induces a homomorphism $\rho_{_{\!N_{\boldsymbol T}(w)}}$ of the polynomial ring $P[z_1, ..., z_m],$ namely, $\rho_{_{\!N_{\boldsymbol T}(w)}}(p) = p\big (N_{\boldsymbol T}(w) \big),\, p\in P[z_1, ..., z_m].$ We study the contractivity and complete contractivity of the homomorphism $\rho_{_{\!N_{\boldsymbol T}(w)}}.$ Starting from the homomorphism $\rho_{_{\!N_{\boldsymbol T}(w)}},$ we construct a natural class of homomorphism $\rho_{_{\!N^{(\lambda)}(w)}}, \lambda>0,$ and relate the properties of $\rho_{_{\!N^{(\lambda)}(w)}}$ to that of $\rho_{_{\!N_{\boldsymbol T}(w)}}.$ Explicit examples arising from the multiplication operators on the Bergman space of $\Omega$ are investigated in detail. Finally, it is shown that contractive properties of $\rho_{_{\!N_{\boldsymbol T}(w)}}$ is equivalent to an inequality for the curvature of the Cowen-Douglas bundle $E_{\boldsymbol T}$.