Numerical range and quasi-sectorial contractions

A method developed in Arlinskiĭ (1987) [1] is applied to study the numerical range of quasi-sectorial contractions and to prove three main results. Our first theorem gives characterization of the maximal sectorial generator A in terms of the corresponding contraction semigroup {exp(−tA)}t⩾0. The second result establishes for these quasi-sectorial contractions a quite accurate localization of their numerical range. We give for this class of semigroups a new proof of the Euler operator-norm approximation: exp(−tA)=limn→∞(I+tA/n)−n, t⩾0, with the optimal estimate: O(1/n), of the convergence rate, which takes into account the value of the sectorial generator angle (the third result).