  Given two Hilbert spaces, $\mathcal{H}$ and $\mathcal{K}$, we introduce an
abstract unitary operator $U$ on $\mathcal{H}$ and its discriminant $T$ on
$\mathcal{K}$ induced by a coisometry from $\mathcal{H}$ to $\mathcal{K}$ and a
unitary involution on $\mathcal{H}$. In a particular case, these operators $U$
and $T$ become the evolution operator of the Szegedy walk on a graph, possibly
infinite, and the transition probability operator thereon. We show the spectral
mapping theorem between $U$ and $T$ via the Joukowsky transform. Using this
result, we have completely detemined the spectrum of the Grover walk on the
Sierpi\'nski lattice, which is pure point and has a Cantor-like structure.
