An alternative proof of the a priori $\tan\Theta$ Theorem

Theoretical and Mathematical Physics 186:1 (2016), 83-92 [English translation]; Teoreticheskaya i Matematicheskaya Fizika 186:1 (2016), 101-112 [Russian original] Let $A$ be a self-adjoint operator in a separable Hilbert space. Suppose that the spectrum of $A$ is formed of two isolated components $\sigma_0$ and $\sigma_1$ such that the set $\sigma_0$ lies in a finite gap of the set $\sigma_1$. Assume that $V$ is a bounded additive self-adjoint perturbation of $A$, off-diagonal with respect to the partition ${\rm spec}(A)=\sigma_0 \cup \sigma_1$. It is known that if $\|V\|