Finite-dimensional perturbations

Abstract:

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