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An alternative proof of the a priori $\tanΘ$ Theorem

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 ${\rm spec}(A)=σ_0 \cup σ_1$. It is known that if $\|V\|<\sqrt{2}{\rm dist}(σ_0,σ_1)$, then the spectrum of the perturbed operator $L=A+V$ consists of two disjoint parts $ω_0$ and $ω_1$ which originate from the corresponding 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: $$\|E_A(σ_0)-E_{L}(ω_0)\|\leq\sin\left(\arctan\frac{\|V\|}{{\rm dist}(σ_0,σ_1)}\right).$$ In the present note, we give a new proof of this bound for $\|V\|<{\rm dist}(σ_0,σ_1)$.