  Despite the intense theoretical and experimental effort, an understanding of
the superconducting pairing mechanism of the high-temperature superconductors
is still lacking. An additional puzzle is the unknown connection between the
superconducting gap and the so-called pseudogap which is a central property of
the most unusual normal state. Angle-resolved photoemission spectroscopy
(ARPES) measurements have revealed a gap-like behavior on parts of the Fermi
surface, leaving a non-gapped segment known as Fermi arc around the diagonal of
the Brillouin zone. Starting from the $t$-$J$ model, in this paper we present a
microscopic approach to investigate physical properties of the pseudogap phase
in the framework of a novel renormalization scheme called PRM. This approach is
based on a stepwise elimination of high-energy transitions using unitary
transformations. We arrive at a renormalized 'free' Hamiltonian for correlated
electrons. The ARPES spectral function along the Fermi surface turns out to be
in good agreement with experiment: We find well-defined excitation peaks around
$\omega=0$ near the nodal direction, which become strongly suppressed around
the antinodal point. The origin of the pseudogap can be traced back to a
suppression of spectral weight from incoherent excitations in a small
$\omega$-range around the Fermi energy. In a subsequent paper, also the
supercunducting phase at moderate hole doping will be discussed within the PRM
approach.
