On the fullness conditions of the eigenvector system of general normal operators

In terms of almosl-pcriodicity of functions $\phi(T(g)x)$ with $T$-uniorder continuous representation of local compact Abel group $G$ of the $(C_0)$ class in weakly full linear topological space $X$, the criterion of fullness for $T$-representation eigenvectors has been proved. In case, when $X$ is reflexive Banach space and $T$ is an isometric representation of group G of $X$ space, with all weighted subspaces having finite dimensions, the existence of full functional system biorthogonal to the union of the basis vectors of weighted snbspace of $T$ representation has been proved.