The Davis–Kahan sin 2Θ theorem in unitarily invariant seminorms

Let A and B be symmetric operators on a finite-dimensional inner product space E, let U be invariant under A and V invariant under B. Suppose that A has two ordered quadratic-form bounds relative to E = U ⊕ Uᗮ:

re ⟪A z, z⟫ ≤ a ‖z‖² on Uᗮ and b ‖y‖² ≤ re ⟪A y, y⟫ on U.

Then the double-angle operator 2 P_{Uᗮ} P_V P_U (Submodule.sinTwoAngleOperator U V), the off-diagonal U → Uᗮ block of 2 P_V, satisfies

(b - a) N(2 P_{Uᗮ} P_V P_U) ≤ 2 N(B - A)

for every unitarily invariant seminorm N. When U and V have the same dimension, the singular values of 2 P_{Uᗮ} P_V P_U are the sines sin 2θᵢ of the doubled principal angles between U and V, so this is Davis's sin 2Θ theorem. Only A is assumed to have a gap; no spectral hypothesis is placed on B, and V is an arbitrary B-invariant subspace. The bound is trivial when b ≤ a. For a one-dimensional V spanned by a unit eigenvector x of B, the double-angle operator has rank at most one and operator norm 2 ‖P_U x‖ ‖P_{Uᗮ} x‖, and the operator-norm case recovers, in finite dimension, the per-eigenvector bound LinearMap.IsSymmetric.sin_two_theta_le.

The argument compares the two Sylvester operators of A and B. Since P_U commutes with A and P_V commutes with B, the operator X = P_{Uᗮ} P_V P_U satisfies

A X - X A = P_{Uᗮ} (P_V H - H P_V) P_U, where H = B - A.

The values of X lie in Uᗮ, where the eigenvalues of A lie in [-‖A‖, a], and X vanishes on Uᗮ, so it only sees the eigenvalues of A on U, which are at least b. The interval/exterior Sylvester bound on invariant subspaces gives (b - a) N(X) ≤ N(A X - X A), and the checkerboard contraction N(H P_V - P_V H) ≤ N(H) bounds the right-hand side by N(H).

Main results

• TauCeti.UnitarilyInvariantSeminorm.mul_map_sinTwoAngleOperator_le_two_mul_map_sub: the sin 2Θ theorem (b - a) N(2 P_{Uᗮ} P_V P_U) ≤ 2 N(B - A).

References

• C. Davis, The rotation of eigenvectors by a perturbation, J. Math. Anal. Appl. 6 (1963), 159–173.
• C. Davis, W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal. 7 (1970), 1–46.

theorem TauCeti.UnitarilyInvariantSeminorm.mul_map_sinTwoAngleOperator_le_two_mul_map_sub {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] (N : UnitarilyInvariantSeminorm 𝕜 E E) {A B : E →L[𝕜] E} (hA : (↑A).IsSymmetric) (hB : (↑B).IsSymmetric) {U V : Submodule 𝕜 E} (hU : ∀ x ∈ U, A x ∈ U) (hV : ∀ x ∈ V, B x ∈ V) {a b : ℝ} (ha : ∀ z ∈ Uᗮ, RCLike.re (inner 𝕜 (A z) z) ≤ a * ‖z‖ ^ 2) (hb : ∀ y ∈ U, b * ‖y‖ ^ 2 ≤ RCLike.re (inner 𝕜 (A y) y)) : (b - a) * N ↑(U.sinTwoAngleOperator V) ≤ 2 * N (↑B - ↑A)

The Davis–Kahan sin 2Θ theorem. Let A and B be symmetric operators on a finite-dimensional inner product space, let U be invariant under A and V invariant under B. If re ⟪A z, z⟫ ≤ a ‖z‖² on Uᗮ and b ‖y‖² ≤ re ⟪A y, y⟫ on U, then the double-angle operator 2 P_{Uᗮ} P_V P_U satisfies (b - a) N(2 P_{Uᗮ} P_V P_U) ≤ 2 N(B - A) for every unitarily invariant seminorm N. No spectral hypothesis is placed on B.