“The concept of generalized inverse in mathematics is a generalization of the notion of inverse of a matrix or function. It is a useful tool to solve systems of linear and nonlinear equations, as well as to study properties of linear or nonlinear operators. There are different types of generalized inverses, according to the type of relationship considered and the conditions imposed on the objects of study. So, if we consider B(H, K) the set of bounded linear operators with H and K be a infinite dimension complex Hilbert spaces, then we define a new generalized inverse, called ϵ-{2,3,4} inverse. This inverse exists even if operator doesn’t has closed range. Therefore, the objectives of this thesis work are: Define a new generalized inverse for bounded linear operators between Banach spaces. Give a block matrix representation of the operator and its generalized inverse. Define a new generalized inverse for bounded linear operators between infinite-dimensional complex Hilbert spaces, where the operators considered does not necessarily have a closed range. Give a block matrix representation of the operator and its generalized inverse. Using block matrix representations of an operator, we will have a detailed structure of its generalized inverse”.