= Solution
For a grid sequence define its <semidiscrete Fourier transform> in the dimensionless frequency variable $\theta\in[-\pi,\pi]$ by
$$
\widehat W(\theta)=\sum_{j\in\mathbb Z}W_je^{-ij\theta},\qquad
W_j=\frac1{2\pi}\int_{-\pi}^{\pi}\widehat W(\theta)e^{ij\theta}d\theta.
$$
For general square-summable data the transform is interpreted as an limit in the <L2 norm>, rather than an <absolutely convergent> pointwise sum. The <Parseval identity> in this normalization is
$$
\|W\|_{\ell_2}^2=\Delta x\sum_{j\in\mathbb Z}|W_j|^2
=\frac{\Delta x}{2\pi}\int_{-\pi}^{\pi}|\widehat W(\theta)|^2d\theta.
$$
The <Fourier symbols> of $D_x^0$ and $D_x^+D_x^-$ are $i\sin\theta/\Delta x$ and $-4\sin^2(\theta/2)/(\Delta x)^2$. Transforming the implicit equation gives
$$
\left(1+4a\mu\sin^2(\theta/2)+i\nu\sin\theta\right)\widehat U^{m+1}=\widehat U^m.
$$
Its <amplification factor> is consequently
$$
G_I(\theta)=\frac1{1+4a\mu\sin^2(\theta/2)+i\nu\sin\theta}.
$$
The denominator has real part at least one because $a\geq0$. It never vanishes, so multiplication by $G_I$ defines a unique next value in the <l2 sequence space>, and
$$
|G_I(\theta)|^2=\frac1{(1+4a\mu\sin^2(\theta/2))^2+\nu^2\sin^2\theta}\leq1.
$$
The <Parseval identity> now gives
$$
\boxed{\|U^{m+1}\|_{\ell_2}\leq\|U^m\|_{\ell_2}\leq\|U^0\|_{\ell_2}\quad\text{for every }m\geq0,\ \Delta x,\Delta t>0.}
$$
This is unconditional <practical stability of a finite difference scheme>: in the course's terminology the numerical evolution does not increase this <norm>. Linearity gives the same estimate for perturbations of the initial data. This contraction convention is stronger than a finite-time mesh-uniform bound allowing physical exponential growth.https://www.numerical.rl.ac.uk/media/people/tyrone-rees/nsde/notes/Lecture12.pdf[Archived Oxford NSDE I Lecture 12, PDF page 2, definition of practical stability as non-growth of the discrete norm]{ref}
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