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 $\mathbf{T}$ in the Cowen-Douglas class ${\mathrm{B}}_1(\Omega ),$ let $N_{\mathbf{T}}(w)$ denote the restriction of $\mathbf{T}$ to the subspace ${\cap }_{i,j=1}^m\ker (T_i-w_{i}I)(T_j-w_{j}I).$ This commuting $m$-tuple $N_{\mathbf{T}}(w)$ of $m+1$ dimensional operators induces a homomorphism ${\rho }_{{}_{N_{\mathbf{T}}(w)}}$ of the polynomial ring $P[z_1,...,z_m],$ namely, ${\rho }_{{}_{N_{\mathbf{T}}(w)}}(p)=p(N_{\mathbf{T}}(w)),p\in P[z_1,...,z_m].$ We study the contractivity and complete contractivity of the homomorphism ${\rho }_{{}_{N_{\mathbf{T}}(w)}}.$ Starting from the homomorphism ${\rho }_{{}_{N_{\mathbf{T}}(w)}},$ 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 ${\rho }_{{}_{N_{\mathbf{T}}(w)}}.$ 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 ${\rho }_{{}_{N_{\mathbf{T}}(w)}}$ is equivalent to an inequality for the curvature of the Cowen-Douglas bundle $E_{\mathbf{T}}$.

arXiv commentThe material in this paper is taken from the PhD thesis, arXiv:1410.6394, of the second author