Vector-spread majorization conjecture for Ritz value errors

Let ${\mathcal X}$ and ${\mathcal Y}$ be subspaces of a Hilbert space ${\mathcal H}$ with ${{\rm \dim}} \mathcal X={{\rm \dim}} \mathcal Y$, and let $A$ be Hermitian. For a vector $v$, write $|v|$ for its componentwise absolute value, and let $\prec_w$ denote weak majorization. Let $P_{\mathcal X}$ and $P_{\mathcal Y}$ be the orthogonal projections onto the two subspaces, let $\Lambda(B)$ denote the vector of eigenvalues of a Hermitian operator $B$, and let ${\rm Spr}_{(\mathcal X+\mathcal Y)}$ be the vector spread defined by the ordered extreme eigenvalues of $(P_{\mathcal X+\mathcal Y}A)|_{\mathcal X+\mathcal Y}$. Then vector-spread majorization conjecture.

$$\left|\Lambda\left((P_{\mathcal X}A)|_{\mathcal X}\right)-\Lambda\left((P_{\mathcal Y}A)|_{\mathcal Y}\right)\right|\prec_w {\rm Spr}_{(\mathcal X+\mathcal Y)}\sin\Theta(\mathcal X,\mathcal Y).$$

If one of the subspaces is $A$-invariant, then

$$\left|\Lambda\left((P_{\mathcal X}A)|_{\mathcal X}\right)-\Lambda\left((P_{\mathcal Y}A)|_{\mathcal Y}\right)\right|\prec_w {\rm Spr}_{(\mathcal X+\mathcal Y)}\sin^2\Theta(\mathcal X,\mathcal Y).$$

Here the products on the right-hand sides are componentwise. The conjecture proposes replacing the scalar spectral-spread factor in existing Rayleigh–Ritz error bounds by the sharper vector of spectral spreads; the authors report that numerical tests motivate both inequalities, which remain unproved in the supplied text.