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

Abstract

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 $\mathrm{spec}(A)={\sigma }_0\cup {\sigma }_1$. It is known that if $\Vert V\Vert <\sqrt{2}\mathrm{dist}({\sigma }_0,{\sigma }_1)$, 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: $$\Vert E_A({\sigma }_0)-E_L({\omega }_0)\Vert \le \sin \left(\arctan \frac{\Vert V\Vert }{\mathrm{dist}({\sigma }_0,{\sigma }_1)}\right).$$ In the present note, we give a new proof of this bound for $\Vert V\Vert <\mathrm{dist}({\sigma }_0,{\sigma }_1)$.

References in corpus (2)

• On the existence of solutions to the operator Riccati equation and the tanΘtheorem
• Bounds on variation of the spectrum and spectral subspaces of a few-body Hamiltonian

Cited by in corpus (1)

• Rayleigh-Ritz majorization error bounds of the mixed type