  This paper continues the previous work (Quantum Inf. Process (2019)) by two
authors of the present paper about a spectral mapping property of chiral
symmetric unitary operators. In physics, they treat non-unitary time-evolution
operators to consider quantum walks in open systems. In this paper, we
generalize the above result to include a chiral symmetric non-unitary operator
whose coin operator only has two eigenvalues. As a result, the spectra of such
non-unitary operators are included in the (possibly non-unit) circle and the
real axis in the complex plane. We also give some examples of our abstract
results, such as non-unitary quantum walks defined by Mochizuki et al.
Moreover, we present an application to the Ihara zeta functions and correlated
random walks on regular graphs, which are not quantum walks.
