On the standard space $l^2$ let us consider the left shift operator $$ L(c_1,c_2,c_3,\ldots)=(c_2,c_3,c_4,\ldots). $$ It is well known that the spectrum of $L$ is the whole unit disk in the complex plane. I would like to approximate $L$ by some sequence of finite-dimensional operators $L_n$. A naive way to do this is to set $L_n$ as follows $$ L_n=\left(\begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ 0 & 0 & 0 & \ddots & 0 \\ 0 & 0 & 0 & \ldots & 1 \\ 0 & 0 & 0 & \ldots & 0 \end{array}\right) $$

However the spectrum of $L_n$ consists only of $0$. Could one suggest more reasonable finite-dimensional approximation sequence $L_n$ such that spectrum of operator $L_n$ gradually fills the unit disk? References are welcome.

Berg's method. It's used for approximating the generators of irrational rotation algebras, one of which can be the shift. $\endgroup$ – David Handelman Nov 19 '16 at 14:14