On reducibility of some operator semigroups and algebras on locally convex spaces (Q2368510)

In the paper under review, some well-known results concerning to reducibility and triangularizability of semigroups and algebras of operators in Banach spaces are extended to the more general setting of locally convex spaces. Among the theorems obtained in this paper, we quote the following: (i) If \({\mathcal S}\) is a family of nuclear operators on a barreled \(H\)-locally convex space, then \({\mathcal S}\) is triangularizable if and only if the trace functional is permutable in \(\mathcal S\); (ii) if \(\mathcal S\) is a semigroup of compact idempotent operators on a barreled locally convex space, then \(\mathcal S\) is triangularizable; (iii) if \(X\) is an infinite-dimensional barreled locally convex space and \(\mathcal S\) is a semigroup of compact operators on \(X\) having the property that the spectrum is submultiplicative in \(\mathcal S\), then \(\mathcal S\) is reducible; (iv) if \(\mathcal S\) is an irreducible semigroup of compact operators on a barreled locally convex space and the spectral radius is submultiplicative in \(\mathcal S\), then it is actually multiplicative.