Ranges of vector states on irreducible operator semigroups (Q343467)

The authors study irreducible semigroups of operators acting, for the most part of the paper, on finite-dimensional Hilbert spaces. Given a non-zero linear functional \(\varphi\) acting on an irreducible semigroup \(\mathcal{S}\), it is known that the range \[ \varphi(\mathcal{S})=\{\varphi(S):S\in\mathcal{S}\} \] of \(\varphi\) must contain at least two elements. The authors initiate a study of possible sets which can arise as ranges of such functionals \(\varphi\), proving in particular that in certain cases the existence of a single non-zero functional \(\varphi\) whose range has exactly two elements already yields valuable information about the structure of the semigroup \(\mathcal{S}\). The authors consider in turn the cases of semigroups of small rank, semigroups of invertible matrices, semigroups of intermediate rank and self-adjoint semigroups. The paper concludes with some remarks concerning semigroups of operators acting on infinite-dimensional Hilbert spaces.