I encountered this problem when studying Spectral Properties of bounded Toeplitz Matrices by Bottcher & Grudsky.
For each polynomial $a=\sum_{n}a_n t^{n}$ which is in Wiener algebra, we define its corresponding infinite Toeplitz matrix, whose $j,k$ entry is:
$$T(a)_{j,k}=a_{j-k}$$
and the finite matrix $T_{N}(a)$ is the upper left $N \times N$ block in $T(a)$. The spectrum of $A$ is given by all $\lambda$ s.t. $A - \lambda I$ is not invertible. The spectral radius of $A$, $rad A$, is given by the largest $|\lambda|$ which is in the spectrum of $A$.
In section 10.3, it is stated that the spectral radius of a finite Toeplitz matrix will not converge to its infinite counterpart, e.g. for $b=t$, $rad T(b)=1$ and $rad T_{N}(b)=0$ for any $N$.
How can this be understood? In particular, I hope to know why $T(b) - \exp(i\theta)I$ is not invertible for some $\theta$.
The matrix is given by
$T_{3}(b)=\begin{pmatrix} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 1 & 0\\ \end{pmatrix}$
and diagonally extends infinitely for the infinite version.
It can however easily been shown that $T(b^{-1})=T(1/t)$ has eigenvalue 1:
$T(1/t)=\begin{pmatrix} 0 & 1 & 0 & \cdots\\ 0 & 0 & 1\\ 0 & 0 & 0\\ \vdots\\ \end{pmatrix}$
since $(1,1,1,1,...)^{T}$ is such an eigenvector. This does not work for $b=t$ though. Could Bottcher & Grudsky be making a mistake here?