The correct answer is $\boxed{{z \over {z – {a^T}}}}$.
The z-transform is a powerful tool for analyzing and designing discrete-time systems. It is defined as
$$F(z) = \sum_{n=-\infty}^{\infty} a_n z^{-n}$$
where $a_n$ is the value of the sequence at time $nT$, and $z$ is a complex number.
The z-transform of the function $f(nT) = a^nT$ is
$$F(z) = \sum_{n=-\infty}^{\infty} a^n z^{-n} = \frac{z}{z-a^T}$$
This can be shown by using the geometric series formula
$$\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$$
Setting $x=a^T$, we get
$$\sum_{n=0}^{\infty} a^{nT} = \frac{1}{1-a^T}$$
Multiplying both sides by $z$ and taking the inverse z-transform, we get
$$F(z) = \frac{z}{z-a^T}$$
Option A is incorrect because it does not take into account the time delay of $nT$. Option B is incorrect because it does not take into account the fact that $a$ is a complex number. Option C is incorrect because it does not take into account the fact that $a^T$ is a complex number. Option D is incorrect because it does not take into account the time delay of $nT$.