On commutativity of the commutant of strongly irreducible operator

Abstract

In 2006, C. L. Jiang and Z. Y. Wang posed an open problem: If T T is a strongly irreducible operator, is A ′ ( T ) / rad A ′ ( T ) {\mathcal {A}}’(T)/\textrm {rad}\,{\mathcal A}’(T) commutative? They conjectured that the answer is positive. In this paper, to negatively answer their problem, a counterexample is given.