Lifting properties in operator ranges

Abstract

Given a bounded positive linear operator A on a Hilbert space H we consider the semi-Hilbertian space (H, <,>_A), where <ℇ, n >_A =< Aℇ,n>. On the other hand, we consider the operator range R(A^1/2) with its canonical Hilbertian structure, denoted by R(A^1/2). In this paper we explore the relationship between different types of operators on (H, <,>_A) with classical subsets of operators on R(A^1/2), like Hermitian, 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.