The spectral operator was introduced by M. L. Lapidus and M. van
Frankenhuijsen [La-vF3] in their reinterpretation of the earlier work of M. L.
Lapidus and H. Maier [LaMa2] on inverse spectral problems and the Riemann
hypothesis. In essence, it is a map that sends the geometry of a fractal string
onto its spectrum. In this survey paper, we present the rigorous functional
analytic framework given by the authors in [HerLa1] and within which to study
the spectral operator. Furthermore, we also give a necessary and sufficient
condition for the invertibility of the spectral operator (in the critical
strip) and therefore obtain a new spectral and operator-theoretic reformulation
of the Riemann hypothesis. More specifically, we show that the spectral
operator is invertible (or equivalently, that zero does not belong to its
spectrum) if and only if the Riemann zeta function zeta(s) does not have any
zeroes on the vertical line Re(s)=c. Hence, it is not invertible in the
mid-fractal case when c=1/2, and it is invertible everywhere else (i.e., for
all c in(0,1) with c not equal to 1/2) if and only if the Riemann hypothesis is
true. We also show the existence of four types of (mathematical) phase
transitions occurring for the spectral operator at the critical fractal
dimension c=1/2 and c=1 concerning the shape of the spectrum, its boundedness,
its invertibility as well as its quasi-invertibility
