Alternative proof of the a priori tan Θ theorem

Abstract

Let A be a self-adjoint operator in a separable Hilbert space. We assume that the spectrum of A consists of two isolated components σ₀ and σ₁ and the set σ₁ is in a finite gap of the set σ₁. It is known that if V is a bounded additive self-adjoint perturbation of A that is off-diagonal with respect to the partition spec(A) = σ₀ ∪ σ₁, then for $\left\| V \right\| < \sqrt 2 d$, where d = dist(σ₀, σ₁), the spectrum of the perturbed operator L = A+V consists of two isolated parts ω₀ and ω₁, which appear as perturbations of the respective spectral sets s0 and s1. Furthermore, we have the sharp upper bound ||EA(σ₀) - EL(ω₀)|| ≤ sin (arctan(||V||/d)) on the difference of the spectral projections E_A(σ₀)) and E_L(ω₀)) corresponding to the spectral sets σ₀ and ω₀ of the operators A and L. We give a new proof of this bound in the case where ||V|| < d.