Isometric and contractive operators in Kreǐn spaces

Let $T$ be a continuous isometric linear operator on a Krein space $\mathcal K$. In general, $T$ is not isometric with respect to a norm on $\mathcal K$ whose metric topology is the Mackey topology on $\mathcal K$. In this note we give a sufficient condition that a norm exists which preserves an isomerty or contraction. We apply this result to prove that, under a certain assumption, the main transformation of a linear system is similar to a Hilbert space contraction. A slight modification of this result is used to give a new proof of a theorem of Davis and Foias. It says that an operator in a Hilbert space is similar to a contraction if a corresponding transfer function is bounded on the open unit disk. As another application it is used to generalize the Beurling-Lax theorem to Krein spaces which are contained continuously and contractively in a space of square summable power series with coefficients in a Krein space.