Lifting properties in operator ranges

Keywords:A-operators, operator ranges. Given a bounded positive linear operator A on a Hilbert spaceH we consider the semi-Hilbertian space (H, 〈 , 〉A), where 〈ξ, η〉A = 〈Aξ, η〉. On the other hand, we consider the operator range R(A1/2) with its canonical Hilbertian structure, denoted by R(A1/2). In this paper we explore the relationship between different types of operators on (H, 〈 , 〉A) with classical subsets of operators on R(A1/2), like Hermi-tian, normal, contractions, projections, partial isometries and so on. We extend a theorem by M. G. Krein on symmetrizable operators and a result by M. Mbekhta on reduced minimum modulus