In this paper we give some algebraic characterizations of an algebraic element with the characteristic polynomial having single roots and then inverstigate the Noether properties of bounded linear operators of the form
\begin{equation} K=\sum\limits_{(i)\in\Gamma}A_{(i)}T^{(i)},\tag{1}\end{equation}
where
$$\Gamma=\left\{(i)=(i_1,i_2,\dots,i_m)\, | \, 0\leq i_k\leq n_k-1, \ k=1,\dots,m\right\}$$ $$A_{(i)}=A_{i_1i_2\dots i_m},\ T^{(i)}=T_1^{i_1}T_2^{i_2}\dots T_m^{i_m},$$
$T_k$ are the commutative algebraic elements of order $n_k$, respectively.