I am considering a self-adjoint semi-infinite Toeplitz matrix, by which I mean $$M = \left(\begin{array}{ccccc} a_0 & a_1 & a_2 & a_3 & \cdots \\ a_1^*& a_0 & a_1 & a_2 & \ddots \\ a_2^* & a_1^*& a_0 & a_1 & \ddots \\ a_3^*& a_2^* & a_1^*& a_0 & \ddots \\ \vdots & \ddots & \ddots & \ddots & \ddots \end{array}\right) $$ where $\{a_n\}_{n=-\infty}^{+\infty}$ are complex numbers, and the star "$^*$" indicates the complex conjugation. It is assumed that $\sum_n \left|a_n\right|$ is finite.
I would like to understand the eigenvalue spectrum of $M$, especially when compared to its infinite analog $M_0$. It is well known (and shown e.g. by going to a Fourier-transformed basis) that the eigenvalues of $$M_0 = \left(\begin{array}{cccccc} \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\ \ddots & a_0 & a_1 & a_2 & a_3 & \ddots \\ \ddots & a_1^*& a_0 & a_1 & a_2 & \ddots \\ \ddots & a_2^* & a_1^*& a_0 & a_1 & \ddots \\ \ddots & a_3^*& a_2^* & a_1^*& a_0 & \ddots \\ \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \end{array}\right) $$ form a one-dimensional family $$\lambda(\omega) = a_0 + 2\sum_{n=1}^{+\infty} \left[\Re[a_n] \cos(n \omega)+\Im[a_n] \sin(n \omega )\right]$$ where $\omega\in [0,2\pi)$. It then follows from the finiteness of $\sum_n |a_n|$ that the spectrum of $M_0$ is both compact and connected, namely an interval $$\textrm{spec}(M_0) = [x_1,x_2]$$ where $|x_{1,2}|<\infty$ are some finite numbers.
However, the spectrum of the semi-infinite matrix $M$ cannot be found using the same Fourier-transform trick, and I suspect there is no exact solution in this case.
The question: Is it true that $\textrm{spec}(M)\subseteq \textrm{spec}(M_0)$?
My simple numerical tests suggest that this should be true, but I didn't manage to prove this and I also failed to find a suitable reference. I was hoping that someone in the Stack Exchange community either readily knows the answer, or could point me to the relevant references/books.
If the answer turns out to be "no", then I also have a second question: Is $\textrm{spec}(M)$ always connected (as is the case for $M_0$), or is it possible that it exhibits some isolated eigenvalues detached from the continuum part of the spectrum?