文摘
Let D denote the open unit disk in the complex plane and let \({\phi}\) denote a holomorphic self-map of D. We show that if \({\phi}\) is not linear-fractional and has its Denjoy-Wolff point on the boundary of D, then 0 is an interior point of the numerical range of the composition operator \({C_{\phi}}\) on the Hardy space \({H^2({\mathbf D})}\). This answers a question raised by Bourdon and Shapiro (Integr Equations Oper Theory 44:410–441, 2002), and, together with their results, provides a complete description of when 0 belongs to the numerical range of a composition operator on \({H^2({\mathbf D})}\).KeywordsNumerical rangeComposition operatorHardy spaceDenjoy-Wolff point