Noncommuting unitary groups and local boundedness

Abstract:

We exhibit two unitary strongly continuous one-parameter groups ${({e^{{A_1}t}})_{t \in {\mathbf {R}}}}$ and ${({e^{{A_2}t}})_{t \in {\mathbf {R}}}}$ acting in a Hilbert space $H$, a dense subspace $D$ of $H$ contained in the domains of ${A_1}$ and ${A_2}$ such that $({A_1}(D) \cup {A_2}(D)) \subset D$ and $({e^{{A_1}t}}(D) \cup {e^{{A_2}t}}(D)) \subset D$ for each $t \in {\mathbf {R}}$, and an element $x$ of $D$ such that the function $t \to \left \| {{A_1}{e^{{A_2}t}}x} \right \|$ is not locally bounded.