Bulletin of the
Korean Mathematical Society BKMS

In this paper, we study some properties of $(\sqrt[k]{H})$ (defined below). In particular we show that an operator $T \in (\sqrt[k]{H})$ satisfying the translation invariant property is hyponormal and an invertible operator $T \in (\sqrt[k]{H})$ and its inverse $T^{-1}$ have a common nontrivial invariant closed set. Also we study some cases which have nontrivial invariant subspaces for an operator in $(\sqrt{H})$.

Keywords: hyponormal operators, hypercyclicity, subscalarity

MSC numbers: 47B20, 47B38