Strong Metrizability for Closed Operators and the Semi-Fredholm Operators between Two Hilbert Spaces

To be able to refine the completion of C(H1, H2), the of set all closed densely defined linear operators between two Hilbert spaces H1 and H2, we define in this paper some new strictly stronger metrics than the gap metric g and we characterize the closure with respect to theses metrics of the subset L(H1, H2) of bounded elements of C(H1, H2). In addition, several operator norm inequalities concerning the equivalence of some metrics on L(H1, H2) are presented. We also establish the semi-Fredholmness and Fredholmness of unbounded in terms of bounded pure contractions.