$L^2$ convergence of this wave equations scheme

Question

I am trying to prove that the scheme $$ \frac{u_j^{n+1}-2u_j^n+u_j^{n-1}}{(\Delta t)^2} + \theta\frac{-u_{j-1}^{n+1}+2u_j^{n+1}-u_{j+1}^{n+1}}{(\Delta x)^2} +\\ (1-2\theta)\frac{-u_{j-1}^{n}+2u_j^{n}-u_{j+1}^{n}}{(\Delta x)^2} +\notag \theta\frac{-u_{j-1}^{n-1}+2u_j^{n-1}-u_{j+1}^{n-1}}{(\Delta x)^2} = 0 $$ is $L^2$ stable; i.e. $\|u^n\|_2\leq C\|u^0\|_2+C\|u^1\|_2$. Applying Fourier Transform we get $$ \hat{u}_k^{n+1} = \frac{ 2 - (1-2\theta)\alpha_k}{1+\theta\alpha_k}\hat{u}_k^n - \hat{u}_k^{n-1} $$ And to study this recurrence We put it in matrix notation: $$ \begin{pmatrix}\hat{u}_k^{n+1}\\ \hat{u}_k^n\end{pmatrix} = \begin{pmatrix}\beta_k & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}\hat{u}_k^n\\ \hat{u}_k^{n-1}\end{pmatrix} $$ where $\beta_k = \frac{ 2 - (1-2\theta)\alpha_k}{1+\theta\alpha_k}$. To this point I am sure the computations are right (because I checked with my reference book, Allaire's Numerical Analysis). So, the scheme is $L^2$ stable iff the matrix $$ A_k = \begin{pmatrix}\beta_k & -1 \\ 1 & 0 \end{pmatrix} $$ is such that $\|A_k\|_2 := \sup_{\|x\|=1}\|A_kx\|_2\leq1$. But, there is known that $\|A_k\|_2$ is the square root of the spectral radius of $A^*A$. The eigenvalues of $A^*A$ are $$ \frac{\beta^2\pm\beta\sqrt{\beta^2+4}}{2}+1$$ And they are $\leq1$ iff $\beta=0$. In Allaire's book it is written that we have stability for $\theta\in[1/4,1/2]$. What's wrong?