On operators in Frechet spaces similar to isometries (Q1179655)

The aim of the paper is the proof of the following theorem: Let \(E\) be a Fréchet space with a barrelled strong dual, \(S\) an isometric mapping of the space \(E\) onto itself and \(T\) such a continuous mapping of the space \(E\) into itself, that for some strongly monotone pseudonorm \(|\cdot|\) on \(E\) defining the topology of \(E\) and for some \(q\in [0,1)\) the condition \(| T^ n x- S^ n x|\geq q| x|\) is valid for all \(x\in E\) with \(| x|\leq \varepsilon\). Then the adjoint operators \(T^*\) and \(S^*\) are similar.