Abstract. Given a bounded positive linear operator $A$ on a Hilbert space ${\cal H}$ we consider the semi-Hilbertian space $({\cal H}, \left\langle, \right\rangle _A)$, where $\left\langle \xi, \eta\right \rangle_A= \left\langle A\xi, \eta\right \rangle $. On the other hand, we consider the operator range $R(A^{1/2})$ with its canonical Hilbertian structure, denoted by ${\bf{R}}(A^{1/2})$. In this paper we explore the relationship between different types of operators on $({\cal H}, \left\langle, \right\rangle _A)$ with classical subsets of operators on ${\bf{R}}(A^{1/2})$, 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.

AMS Subject Classification (1991): 46C05, 47A05, 47A30

Keyword(s): A, -operators, operator ranges

Received July 11, 2008. (Registered under 6432/2009.)