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 ( − t A ) } 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 ( − t A ) = lim n → ∞ ( I + t A / 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).