An alternative proof of the a priori $\tan \mathrm{\Theta }$ 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 ${\sigma }_0$ 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 ${\rm spec}(A)=\sigma_0 \cup \sigma_1$\mathrm{.}Itisknownthatif$|V|<\sqrt{2}{\rm dist}(\sigma_0,\sigma_1)$, then the spectrum of the perturbed operator $L=A+V$ consists of two disjoint parts ${\omega }_0$ and ${\omega }_1$ which originate from the corresponding initial spectral subsets ${\sigma }_0$ and ${\sigma }_1$. Moreover, for the difference of the spectral projections $E_A({\sigma }_0)$ and $E_L({\omega }_0)$ of $A$ and $L$ associated with the spectral sets ${\sigma }_0$ and ${\omega }_0$, respectively, the following sharp norm bound holds: $$\mathrm{\parallel }{E}_A({\sigma }_0)-E_L({\omega }_0)\mathrm{\parallel }\le \sin (\arctan \frac{\mathrm{\parallel }V\mathrm{\parallel }}{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}({\sigma }_0,{\sigma }_1)})\mathrm{.}$$ In the present note, we give a new proof of this bound for $\mathrm{\parallel }V\mathrm{\parallel }<\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}({\sigma }_0,{\sigma }_1)$.