We report on a detailed spectroscopic study of 12 very massive and luminous stars (M 35M?) in the core of the compact cluster R136a, near the center of the 30 Doradus complex. The three brightest stars of the cluster, R136a1, R136a2, and R136a3, have been investigated earlier by de Koter, Heap, & Hubeny. Low-resolution spectra (<200 km s-1) of the program stars were obtained with the GHRS and FOS spectrographs on the Hubble Space Telescope. These instruments covered the spectral range from 1200 to 1750 ? and from 3200 to 6700 ?, respectively. Fundamental stellar parameters were obtained by fitting the observations by model spectra calculated with the unified ISA-WIND code of de Koter et al. supplemented by synthetic data calculated using the program TLUSTY. We find that the stars are almost exclusively of spectral type O3. They occupy only a relatively narrow range in effective temperatures between 40 and 46 kK. The reason for these similar Teff's is that the isochrone of these very massive stars, which we determined to be at ~2 Myr, runs almost vertically in the H-R diagram. We present a quantitative method of determining the effective temperature of O3-type stars based on the strength of the O V ?1371 line. Present-day evolutionary calculations by Meynet et al. imply that the program stars have initial masses in the range of Mi ~ 37-76 M?. The observed mass-loss rates are up to 3 (2) times higher than is assumed in these evolution tracks when adopting a metallicity Z = 0.004 (0.008) for the LMC. The high observed mass-loss rates imply that already at an age of ~2 Myr the most luminous of our program stars will have lost a significant fraction of their respective initial masses. For the least luminous stars investigated in this paper, the observed mass loss agrees with the prediction by the theory of radiation-driven winds (Kudritzki et al.). However, for increasing luminosity the observed mass loss becomes larger, reaching up to 3-4 times what is expected from the theory. Such an increasing discrepancy fits in with the results of de Koter et al., where an observed overpredicted mass-loss ratio of up to 8 was reported for the brightest members of the R136a cluster, for which Mi ~ 100 M? was found. The failure of the theory is also present when one compares observed over predicted wind momentum as a function of wind performance number. This strongly indicates that the shortcoming of the present state of the theory is connected to the neglect of effects of multiple photon momentum transfer.