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

TL;DR

This paper investigates curvature inequalities for operators in the Cowen-Douglas class on planar domains, establishing the uniqueness of extremal operators and characterizing which bundle shifts are extremal in multiply connected domains.

Contribution

It proves the uniqueness of extremal operators in the Cowen-Douglas class related to curvature bounds and explicitly characterizes extremal bundle shifts in multiply connected domains.

Findings

Extremal operators are uniquely determined by their curvature at a fixed point.

Only some rank 1 bundle shifts serve as extremal operators in multiply connected domains.

Explicit descriptions of extremal bundle shifts are provided.

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 $\mathcal K_T(w) \leq - 4 \pi^2 S_\Omega(w,w)^2,$ where $S_\Omega$ is the S\"{z}ego 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 $\mathcal K_{S^*}(w) = 4 \pi^2 S_\Omega(w,w)^2.$ 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.