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 ${cap_{i,j=1}^mker(T_i-w_iI)(T_j-w_jI)}.$ 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, $rho_{_{!N_{boldsymbol T}(w)}}(p) = pbig (N_{boldsymbol T}(w) big),, pin P[z_1, ..., z_m].$ 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 $E_{boldsymbol T}$.
تحميل البحث