  The notion of a K\"ahler structure for a differential calculus was recently
introduced by the second author as a framework in which to study the
noncommutative geometry of the quantum flag manifolds. It was subsequently
shown that any covariant positive definite K\"ahler structure has a canonically
associated triple satisfying, up to the compact resolvent condition, Connes'
axioms for a spectral triple. In this paper we begin the development of a
robust framework in which to investigate the compact resolvent condition, and
moreover, the general spectral behaviour of covariant K\"ahler structures. This
framework is then applied to quantum projective space endowed with its
Heckenberger-Kolb differential calculus. An even spectral triple with
non-trivial associated K-homology class is produced, directly q-deforming the
Dirac-Dolbeault operator of complex projective space. Finally, the extension of
this approach to a certain canonical larger class of compact quantum Hermitian
symmetric spaces is discussed in detail.
