2026/07/23 by Raul E. Curto, Thankarajan Prasad
#math.FA
In this paper we prove that a square root of a cyclic normal operator is unitarily equivalent to a block multiplication operator on a vector-valued Lebesgue space with a 2--normal symbol. In addition, we show that a cyclic operator which admits a cyclic 2--normal extension is unitarily equivalent to a block multiplication operator on a corresponding vector-valued Hardy space with 2--normal symbol. We consider the existence of bounded point evaluations in the vector-valued Hardy space setting; as an application, we prove that an operator admitting a cyclic 2--normal extension has nontrivial invariant subspaces. We also study uniqueness for minimal n--normal extensions of operators that have cyclic 2--normal extensions.