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

TL;DR

This paper presents a new proof of a sharp bound on the difference of spectral projections for a self-adjoint operator under off-diagonal perturbations, refining the understanding of spectral stability in operator theory.

Contribution

It provides an alternative proof of the $ an heta$ theorem with improved clarity and potentially broader applicability in spectral perturbation analysis.

Findings

Established a sharp norm bound for spectral projection differences.

Provided a new proof technique for the $ an heta$ theorem.

Confirmed the bound holds under specific spectral gap conditions.

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 ${\rm spec}(A)=\sigma_0 \cup \sigma_1$. It is known that if $\|V\|<\sqrt{2}{\rm dist}(\sigma_0,\sigma_1)$, 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: $$\|E_A(\sigma_0)-E_{L}(\omega_0)\|\leq\sin\left(\arctan\frac{\|V\|}{{\rm dist}(\sigma_0,\sigma_1)}\right).$$ In the present note, we give a new proof of this bound for $\|V\|<{\rm dist}(\sigma_0,\sigma_1)$.