Alternative proof of the a priori tan Θ theorem

A. K. Motovilov
2016 Theoretical and mathematical physics  
Let A be a self-adjoint operator in a separable Hilbert space. Suppose that the spectrum of A is formed of two isolated components σ_0 and σ_1 such that the set σ_0 lies in a finite gap of the set σ_1. Assume that V is a bounded additive self-adjoint perturbation of A, off-diagonal with respect to the partition spec(A)=σ_0 ∪σ_1. It is known that if V<√(2) dist(σ_0,σ_1), then the spectrum of the perturbed operator L=A+V consists of two disjoint parts ω_0 and ω_1 which originate from the
more » ... ding initial spectral subsets σ_0 and σ_1. Moreover, for the difference of the spectral projections E_A(σ_0) and E_L(ω_0) of A and L associated with the spectral sets σ_0 and ω_0, respectively, the following sharp norm bound holds: E_A(σ_0)-E_L(ω_0)≤sin(arctanV/ dist(σ_0,σ_1)). In the present note, we give a new proof of this bound for V< dist(σ_0,σ_1).
doi:10.1134/s0040577916010074 fatcat:vxpfv2nzdzf2zamypcc23srgva