  The generalized spectral theory is an effective approach to analyze a linear
operator on a Hilbert space $\mathcal{H}$ with a continuous spectrum. The
generalized spectrum is computed via analytic continuations of the resolvent
operators using a dense locally convex subspace $X$ of $\mathcal{H}$ and its
dual space $X'$. The three topological spaces $X \subset \mathcal{H} \subset
X'$ is called the rigged Hilbert space or the Gelfand triplet. In this paper,
the generalized spectra of the Perron-Frobenius operators of the one-sided and
two-sided shifts of finite types (symbolic dynamical systems) are determined. A
one-sided subshift of finite type which is conjugate to the multiplication with
the golden ration on $[0,1]$ modulo $1$ is also considered. A new construction
of the Gelfand triplet for the generalized spectrum of symbolic dynamical
systems is proposed by means of an algebraic procedure. The asymptotic formula
of the iteration of Perron-Frobenius operators is also given. The iteration
converges to the mixing state whose rate of convergence is determined by the
generalized spectrum.
