This article deals with semigroups of bounded operators in a complex Banach space \(X\); these semigroups are defined and analytic on an open sector \(\Sigma= \{se^{i\phi}+ te^{i\psi}: s,t> 0\}\) in the complex plain \(\mathbb{C}\) and describe solutions of nonlinear evolution equations of type \[ {d\over d\xi} u(\xi)= Au(\xi)\quad (\xi\in \Sigma\subset\mathbb{C}). \] Under natural conditions each semigroup \(W\) of this type generates a pair of boundary semigroups \(S\) and \(T\) defined on the boundary rays of \(\Sigma\).
The main result of the article presents a characterization of \(W\) in terms of \(S\) and \(T\); in particular, it is described conditions under which there exists an analytical semigroup \(W\) with given boundary semigroups \(S\) and \(T\).