Rayleigh–Ritz mixed-type majorization error bounds conjecture

Let $A$ be a Hermitian matrix, and let the columns of matrices $X$ and $Y$ form orthonormal bases for subspaces $d4fd$ and $d4fe$ of equal dimension. Write $P_{\mathcal X}$ and $P_{\mathcal Y}$ for the corresponding orthogonal projectors, let $R_X=AX-X\rho(X)$ and $R_Y=AY-Y\rho(Y)$ be the residual matrices, and let $\Lambda(\cdot)$ and $S(\cdot)$ denote the vectors of eigenvalues and singular values, respectively. Assume that the largest principal angle $\Theta(\mathcal X,\mathcal Y)$ satisfies $\Theta(\mathcal X,\mathcal Y)<\pi/2$. Then

Rayleigh–Ritz mixed-type majorization error bounds conjecture. The following weak majorization inequalities hold:

$$\left|\Lambda\left(X^HAX\right)-\Lambda\left(Y^HAY\right)\right|\prec_w \frac{S\left(P_{\mathcal Y}R_X\right)+S\left(P_{\mathcal X}R_Y\right)}{\cos\left(\Theta\left(\mathcal X,\mathcal Y\right)\right)},$$ $$\left|\Lambda\left(X^HAX\right)-\Lambda\left(Y^HAY\right)\right|\prec_w \left\{S\left(P_{\mathcal X+\mathcal Y}R_X\right)+S\left(P_{\mathcal X+\mathcal Y}R_Y\right)\right\}\tan\left(\Theta\left(\mathcal X,\mathcal Y\right)\right).$$

If $\mathcal X$ is $A$-invariant, then

$$\left|\Lambda\left(X^HAX\right)-\Lambda\left(Y^HAY\right)\right|\prec_w \frac{S\left(P_{\mathcal X}R_Y\right)}{\cos\left(\Theta\left(\mathcal X,\mathcal Y\right)\right)},$$ $$\left|\Lambda\left(X^HAX\right)-\Lambda\left(Y^HAY\right)\right|\prec_w S\left(P_{\mathcal X+\mathcal Y}R_Y\right)\tan\left(\Theta\left(\mathcal X,\mathcal Y\right)\right).$$

These bounds would generalize scalar Rayleigh-quotient error estimates to equal-dimensional subspaces and provide majorization bounds for changes in Ritz values, including approximation-error bounds when one subspace is invariant.