On closed range operators and their characterization via p-Schatten ideals

Abstract

Let S and T be two closed-range bounded linear operators defined on a complex Hilbert space H . The main objective of this paper is to investigate the conditions under which the equality involving the p-Schatten norm, a specific case of symmetric or unitarily invariant norms, holds: S∗XT† +S†XT∗p = 2PXQp, for any X ∈ B(H ), where P = SS† and Q = T†T with 1 < p < ∞. Here, S† and T† denote the Moore-Penrose inverses of S and T , respectively.