Finite-dimensional perturbations

Abstract

Let A be a normal operator on the Hilbert space H \mathcal {H} and let B be an operator of finite rank, rank B = m B = m , such that A + B A + B is normal. Moreover let E (resp. F) denote the spectral projections of A (resp. A + B A + B ) for the set { ζ ∈ C | | ζ − λ | ⩽ α } \{ \zeta \in {\mathbf {C}}||\zeta - \lambda | \leqslant \alpha \} . Then dim E − m ⩽ dim ⁡ F ⩽ dim ⁡ E + m \dim \;E - m \leqslant \dim F \leqslant \dim E + m .