Exercise 7.4.10

Since $x\in C_y$, we may assume $x=T^m(y)$ for some integer $m$. If $\varphi (t)$ is the $T$-annihilator of $x$ and $p(t)$ is the $T$-annihilator of $y$, then we know $p(T)(x)=p(T)(T^m(y))=T^m(p(T)(y))=0$. Hence $p(t)$ is a factor of $\varphi (t)$. If $x=0$, then we have $p(t)=1$ and $y=0$. The statement is true for this case. So we assume that $x\ne 0$. Thus we know that $y\ne 0$ otherwise $x$ is zero. Hence we know $p(t)=\varphi (t)$. By Exercise 7.3.15, the dimension of $C_x$ is equal to the dimension of $C_y$. Since $x=T^m(y)$ we know that $C_x\subset C_y$. Finally we know that they are the same since they have the same dimension.