Knowledge about the spectrum of the Uzawa pressure operator is important for solving and performing an error analysis of the Stokes problem. The infimum of the spectrum of the Uzawa pressure operator is significant, for instance, it gives information about the rate of convergence of numerical methods for the Stokes problem. The spectrum of the Uzawa pressure operator is still not known for the case of a square domain. This thesis provides some results related to this problem. It depicts efforts made in estimating the infimum of the spectrum of the Uzawa pressure operator. In 1996, M. Gaultier and M. Lezaun gave an upper bound equal to 0.2260 for the infimum of the spectrum of the Uzawa pressure operator. We have improved it to 0.20164. We conclude this thesis by giving a conjecture that the infimum of the spectrum of the Uzawa pressure operator is equal to 0.181690.