Infinitely divisible metrics and curvature inequalities for operators in the Cowen-Douglas class (Q2869838)

Let \(B_1(\mathbb D)\) be the class of operators introduced by \textit{M. J. Cowen} and \textit{R. G. Douglas} [Acta Math. 141, 187--261 (1978; Zbl 0427.47016)] and let \(\mathcal{K}_T\) denote the curvature of an operator \(B_1(\mathbb D)\). A special case of the curvature inequality proved by \textit{G. Misra} [J. Oper. Theory 11, 305--317 (1984; Zbl 0544.47015)] says that, if \(T\in B_1(\mathbb D)\) is a contraction, then \(\mathcal{K}_T(w)\leq\mathcal{K}_{S^\ast}(w)\) for all \(w\in\mathbb D\), where \(S^\ast\) is the backward shift operator. First, the authors give an explicit example of a non-contractive operator \(T\) satisfying the same curvature inequality. Further, they study additional conditions on the curvature, which ensure contractivity.