The local reduced minimum modulus on a Hilbert space

Let H be a complex Hilbert space and let \({\mathcal {B}}(H)\) be the algebra of all bounded linear operators on H. In this paper, for \(T\in {\mathcal {B}}(H)\) and a unit vector \(x\in H\), we introduce a local version of the reduced minimum modulus of T at x, noted by \(\gamma (T, x)\). Properties of this quantity are investigated. We study the relations between \(\gamma (T, x)\) and the Moore–Penrose inverse, spectrum of \(\vert T\vert \) and the local spectrum of \(\vert T\vert \) at x. At the end of this paper we will be interested in several problems around this quantity (preserving, continuity, local spectral theory).