(A, m)-partial isometries in semi-Hilbertian spaces

Abstract

In this paper, we introduce the notion of -partial isometry for a positive operator A and a nonnegative integer m. This family of operators contains both the class of -isometries discussed in Sid Ahmed and Saddi [A-m-isometric operators in semi-Hilbertian spaces. Linear Algebra Appl. 2012;436:3930–3942] and that of m-partial isometries introduced in Saddi and Sid Ahmed [m-partial isometries on Hilbert spaces. Int J Funct Anal Oper Theory Appl. 2010;2(1):67–83]. First, we give some interesting algebraic properties of -partial isometries, then we discuss a necessary and sufficient condition for an -partial isometry to be an -isometry. Finally, we give some spectral properties of -partial isometries.