The a priori tan Θ theorem for spectral subspaces

Abstract

Let A be a self-adjoint operator on a separable Hilbert space ${{\mathfrak{H}}}$ . Assume that the spectrum of A consists of two disjoint components σ ₀ and σ ₁ such that the set σ ₀ lies in a finite gap of the set σ ₁. Let V be a bounded self-adjoint operator on ${{\mathfrak{H}}}$ off-diagonal with respect to the partition ${{\rm spec}(A)=\sigma_0\cup\sigma_1}$ . It is known that if ${\|V\| < \sqrt{2}d}$ , where ${d={\mathop{\rm dist}}(\sigma_0,\sigma_1)}$ , then the perturbation V does not close the gaps between σ ₀ and σ ₁ and the spectrum of the perturbed operator L = A + V consists of two isolated components ω₀ and ω₁ originating from σ ₀ and σ ₁, respectively. Furthermore, it is known that if V satisfies the stronger bound ${\|V\| < d}$ then for the difference of the spectral projections ${{\sf E}_A(\sigma_0)}$ and ${{\sf E}_{L}(\omega_0)}$ of A and L associated with the spectral sets σ ₀ and ω₀, respectively, the following sharp norm estimate holds:

$$\|{\sf E}_A(\sigma_0)-{\sf E}_{L}(\omega_0)\| \leq\sin\left(\arctan\frac{\|V\|}{d}\right).$$

In the present work we prove that this estimate remains valid and sharp also for   ${d\leq \|V\| < \sqrt{2}d}$ , which completely settles the issue.