Phases of Sectorial Operators

In this paper, we first define the phases of a sectorial operator based on the numerical range. We are interested in the estimation of the spectrum phases of AB based on the phases of two sectorial operators A and B. Motivated by the classical small gain theorem, we formulate an operator small phase theorem with necessity for the invertibility of \(I+AB\), which plays a crucial role in feedback stability analysis. Afterwards, we consider the special class of sectorial operators of the form \(P+K\), where P is strictly positive and K is compact. More properties of the phases for those operators are studied, including those of compressions, Schur complements, operator means and products. Finally, for the special class of sectorial operators, we further establish a majorization relation between the phases of the spectrum of AB and the phases of two operators A and B.