An alternative proof of the a priori Θ Theorem

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Let A be a self-adjoint operator in a separable Hilbert space. Suppose that the spectrum of A is formed of two isolated components σ₀ and σ₁ such that the set σ₀ lies in a finite gap of the set σ₁. Assume that V is a bounded additive self-adjoint perturbation of A, off-diagonal with respect to the partition spec(A)=σ₀ ∪ σ₁. It is known that if V<√2 dist(σ₀,σ₁), then the spectrum of the perturbed operator L=A+V consists of two disjoint parts ω₀ and ω₁ which originate from the corresponding initial spectral subsets σ₀ and σ₁. Moreover, for the difference of the spectral projections Eₐ(σ₀) and Eₗ(ω₀) of A and L associated with the spectral sets σ₀ and ω₀, respectively, the following sharp norm bound holds: Eₐ(σ₀)-Eₗ(ω₀)≤(V dist(σ₀,σ₁)). In the present note, we give a new proof of this bound for V< dist(σ₀,σ₁).