Curvature inequalities for operators in the Cowen-Douglas class of a planar domain

Abstract

Fix a bounded planar domain $\Omega .$ If an operator $T,$ in the Cowen-Douglas class $B_1(\Omega ),$ admits the compact set $\bar{\Omega }$ as a spectral set, then the curvature inequality ${\mathcal{K}}_T(w)\le -4{\pi }^2{S}_{\Omega }(w,w{)}^2,$ where $S_{\Omega }$ is the Szego kernel of the domain $\Omega ,$ is evident. Except when $\Omega$ is simply connected, the existence of an operator for which ${\mathcal{K}}_T(w)=4{\pi }^2{S}_{\Omega }(w,w{)}^2$ for all $w$ in $\Omega$ is not known. However, one knows that if $w$ is a fixed but arbitrary point in $\Omega ,$ then there exists a bundle shift of rank $1,$ say $S,$ depending on this $w,$ such that ${\mathcal{K}}_{S^{\ast }}(w)=4{\pi }^2{S}_{\Omega }(w,w{)}^2.$ We prove that these {\em extremal} operators are uniquely determined: If $T_1$ and $T_2$ are two operators in $B_1(\Omega )$ each of which is the adjoint of a rank $1$ bundle shift and ${\mathcal{K}}_{T_1}(w)=-4{\pi }^{2}S(w,w{)}^2={\mathcal{K}}_{T_2}(w)$ for a fixed $w$ in $\Omega ,$ then $T_1$ and $T_2$ are unitarily equivalent. A surprising consequence is that the adjoint of only some of the bundle shifts of rank $1$ occur as extremal operators in domains of connectivity greater than $1.$ These are described explicitly.

18 pages