Quasi-isometries in semi-Hilbertian spaces

The concept of quasi-isometry on a Hilbert space H studied by Patel [S.M. Patel, A note on quasi-isometries, Glas. Mat. 35(55) (2000) 307–312; S.M. Patel, A note on quasi-isometries II, Glas. Mat. 38(58) (2003) 111–120] is generalized in the context of A-contractions T (i.e. T∗AT⩽A), A⩾0 and T being bounded linear operators on H. In fact, the new concept is related on the semi-inner product induced by A on H. Other results on operator ranges and invariant null-subspaces for certain A-contractions are obtained.