On an isometric representation with the maximal set of spectral subspaces

It was proved the theorem. Let $G$ be a locally compact noncompact separable Abelian group. Then there exists an isometric representation of the group $G$ in a Banach space $X$ without eigenvectors for which any spectral subspace $L(K)\ne\{0\}$ if $K$ contains a nonempty perfect subset.