Lifting properties in operator ranges

MARÍA LAURA ARIAS, GUSTAVO CORACH, MARIA CELESTE GONZALEZ

ACTA SCIENTIARUM MATHEMATICARUM (SZEGED)

BOLYAI INSTITUTE, UNIVERSITY OF SZEGED

Lugar: Budapest; Año: 2009 vol. 75 p. 635 - 653

0001-6969

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