Sectorial operators and normalized numerical range Original Research Article

We study the notion of sectorial operator in a Hilbert space. According to the classical definition, the numerical range View the MathML source of a sectorial operator A is contained in a sector View the MathML source, and this is equivalent to a certain inverse estimate valid outside View the MathML source. In this paper we show that the validity of the same estimate, but with a factor >1, is equivalent to the validity of a certain strengthened Cauchy–Schwarz inequality for all pairs w,Aw. This extends the original characterization in terms of View the MathML source by a more general characterization based on a normalized numerical range View the MathML source. We also show how View the MathML source can be computed numerically.