An irreducible semigroup of non-negative square-zero operators
An irreducible multiplicative semigroup of non-negative square-zero operators acting on \(L^p[0,1)\) for \(1\leq p<\infty\) illustrates the complexity of the structure possible in such seemingly simple objects as semigroups of square-zero operators. Recently, a number of positive results have been obtained concerning reducibility of semigroups of (possibly non-negative) compact operators. In the authors' opinion, the semigroup constructed or a slightly modified version of it provides an almost universal counterexample to show that very few of these results can be extended to general (possibly non-negative) operator semigroups. The example proposed in the paper provides the first example of an irreducible semigroup of non-negative nilpotent operators where the order of nilpotency is bounded. This answers an open question of \textit{Y. Zhong} [Trans. Am. Math. Soc. 347, 3093-3100 (1995; Zbl 0846.47002)].
scientific article; zbMATH DE number 1754750