Fan dominance and the ideal inequalities

For linear maps A C : E →ₗ[𝕜] F between finite-dimensional inner product spaces, this file proves that A lies in the convex hull of the two-sided unitary orbit {U C V} of C exactly when every Ky Fan sum of A is at most the corresponding Ky Fan sum of C:

A ∈ conv {U C V} ↔ ∀ k, σ₀(A) + ⋯ + σₖ₋₁(A) ≤ σ₀(C) + ⋯ + σₖ₋₁(C).

Every unitarily invariant seminorm is bounded on that hull by its value at C, so this gives Fan dominance: if the Ky Fan sums of A are dominated by those of C, then N A ≤ N C for every unitarily invariant seminorm N. A single family of Ky Fan estimates thus yields the corresponding estimates in the operator, Frobenius, Ky Fan and nuclear norms at once.

The hard direction reduces to the transfer descent for weak majorization (TauCeti.IsSymmetricConvex.mem_of_prefixSum_le). With p = min (dim E) (dim F), write C = ∑ᵢ σᵢ(C) wᵢ ⊗ vᵢ over orthonormal families (vᵢ)_{i < p} and (wᵢ)_{i < p} (LinearMap.exists_orthonormal_eq_sum_singularValues_smul_rankOne). The tuples x : Fin p → ℝ with ∑ᵢ xᵢ wᵢ ⊗ vᵢ in the hull form a symmetric convex set: permuting or changing the sign of the coordinates of x amounts to permuting or changing the sign of the vectors wᵢ and vᵢ, which is realized by unitaries of F and E. This set contains the singular values of C, so it contains those of A, which are weakly majorized by them; and A itself is a unitary multiple of ∑ᵢ σᵢ(A) wᵢ ⊗ vᵢ.

Consequences follow. If ‖B x‖ ≤ c ‖A x‖ for every x, then σᵢ(B) ≤ c σᵢ(A), so Fan dominance gives the pointwise comparison N B ≤ c N A. Taking B = C A gives the ideal inequalities N (C A) ≤ ‖C‖ N A and, through adjoints, N (A C) ≤ ‖C‖ N A; in particular orthogonal projections do not increase N. Applied to the inverse of a map C bounded below by c > 0, they give the reverse inequalities c N A ≤ N (C A) and c N A ≤ N (A C). For a symmetric right factor C and a map A vanishing on Tᗮ, the right ideal inequalities only need the bounds on C along T, since (A C)† = C A† and A† takes values in T.

Main results

• LinearMap.mem_twoSidedUnitaryOrbitHull_iff_kyFanSum_le: the orbit-hull characterization of Ky Fan domination.
• TauCeti.UnitarilyInvariantSeminorm.map_le_of_kyFanSum_le: Fan dominance.
• TauCeti.UnitarilyInvariantSeminorm.map_le_mul_of_norm_apply_le, TauCeti.UnitarilyInvariantSeminorm.mul_map_le_of_mul_norm_apply_le: the pointwise comparison and its reverse.
• TauCeti.UnitarilyInvariantSeminorm.map_comp_le_left, TauCeti.UnitarilyInvariantSeminorm.map_comp_le_right: the ideal inequalities.
• TauCeti.UnitarilyInvariantSeminorm.mul_map_le_map_comp_left, TauCeti.UnitarilyInvariantSeminorm.mul_map_le_map_comp_right: the reverse ideal inequalities for factors bounded below.
• TauCeti.UnitarilyInvariantSeminorm.map_comp_le_right_of_isSymmetric, TauCeti.UnitarilyInvariantSeminorm.mul_map_le_map_comp_right_of_isSymmetric: the right ideal inequalities for a symmetric factor bounded, or bounded below, on a subspace.

References

• K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 760–766.
• R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics 169, Springer, 1997, Theorem IV.2.2 (Fan dominance) and Proposition IV.2.4 (ideal inequalities).
• L. Mirsky, Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. Oxford Ser. (2) 11 (1960), 50–59.

theorem LinearMap.mem_twoSidedUnitaryOrbitHull_iff_kyFanSum_le {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] {A C : E →ₗ[𝕜] F} : A ∈ C.twoSidedUnitaryOrbitHull ↔ ∀ (k : ℕ), A.kyFanSum k ≤ C.kyFanSum k

Orbit-hull characterization of Ky Fan domination. A map A lies in the convex hull of the two-sided unitary orbit of C exactly when every Ky Fan sum of A is at most the corresponding Ky Fan sum of C.

theorem TauCeti.UnitarilyInvariantSeminorm.map_le_of_kyFanSum_le {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 F] (N : UnitarilyInvariantSeminorm 𝕜 E F) {A C : E →ₗ[𝕜] F} (h : ∀ (k : ℕ), A.kyFanSum k ≤ C.kyFanSum k) : N A ≤ N C

Fan dominance. If every Ky Fan sum of A is at most the corresponding Ky Fan sum of C, then N A ≤ N C for every unitarily invariant seminorm N.

Pointwise comparison

theorem TauCeti.UnitarilyInvariantSeminorm.map_le_mul_of_norm_apply_le {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 F] (N : UnitarilyInvariantSeminorm 𝕜 E F) {A B : E →ₗ[𝕜] F} {c : ℝ} (h : ∀ (x : E), ‖B x‖ ≤ c * ‖A x‖) : N B ≤ c * N A

Pointwise comparison. If ‖B x‖ ≤ c * ‖A x‖ for every x, then N B ≤ c * N A for every unitarily invariant seminorm N.

theorem TauCeti.UnitarilyInvariantSeminorm.mul_map_le_of_mul_norm_apply_le {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 F] (N : UnitarilyInvariantSeminorm 𝕜 E F) {A B : E →ₗ[𝕜] F} {c : ℝ} (h : ∀ (x : E), c * ‖A x‖ ≤ ‖B x‖) : c * N A ≤ N B

Reverse pointwise comparison. If c * ‖A x‖ ≤ ‖B x‖ for every x, then c * N A ≤ N B for every unitarily invariant seminorm N.

Ideal inequalities

theorem TauCeti.UnitarilyInvariantSeminorm.map_comp_le_left {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 F] (N : UnitarilyInvariantSeminorm 𝕜 E F) (C : F →ₗ[𝕜] F) (A : E →ₗ[𝕜] F) {c : ℝ} (hC : ∀ (y : F), ‖C y‖ ≤ c * ‖y‖) : N (C ∘ₗ A) ≤ c * N A

Ideal inequality, left factor. If ‖C y‖ ≤ c * ‖y‖ for every y (for c ≥ 0: if C has operator norm at most c), then N (C A) ≤ c * N A for every unitarily invariant seminorm N.

theorem TauCeti.UnitarilyInvariantSeminorm.map_comp_le_right {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 F] (N : UnitarilyInvariantSeminorm 𝕜 E F) (A : E →ₗ[𝕜] F) (C : E →ₗ[𝕜] E) {c : ℝ} (hC : ∀ (x : E), ‖C x‖ ≤ c * ‖x‖) : N (A ∘ₗ C) ≤ c * N A

Ideal inequality, right factor. If ‖C x‖ ≤ c * ‖x‖ for every x (for c ≥ 0: if C has operator norm at most c), then N (A C) ≤ c * N A for every unitarily invariant seminorm N.

theorem TauCeti.UnitarilyInvariantSeminorm.mul_map_le_map_comp_left {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 F] (N : UnitarilyInvariantSeminorm 𝕜 E F) (C : F →ₗ[𝕜] F) (A : E →ₗ[𝕜] F) {c : ℝ} (hC : ∀ (y : F), c * ‖y‖ ≤ ‖C y‖) : c * N A ≤ N (C ∘ₗ A)

Reverse ideal inequality, left factor. If c * ‖y‖ ≤ ‖C y‖ for every y, then c * N A ≤ N (C A) for every unitarily invariant seminorm N.

theorem TauCeti.UnitarilyInvariantSeminorm.mul_map_le_map_comp_right {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 F] (N : UnitarilyInvariantSeminorm 𝕜 E F) (A : E →ₗ[𝕜] F) (C : E →ₗ[𝕜] E) {c : ℝ} (hC : ∀ (x : E), c * ‖x‖ ≤ ‖C x‖) : c * N A ≤ N (A ∘ₗ C)

Reverse ideal inequality, right factor. If c * ‖x‖ ≤ ‖C x‖ for every x, then c * N A ≤ N (A C) for every unitarily invariant seminorm N.

theorem TauCeti.UnitarilyInvariantSeminorm.map_starProjection_comp_le {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 F] (N : UnitarilyInvariantSeminorm 𝕜 E F) (U : Submodule 𝕜 F) (A : E →L[𝕜] F) : N ↑(U.starProjection ∘SL A) ≤ N ↑A

Composing with an orthogonal projection on the left does not increase a unitarily invariant seminorm.

theorem TauCeti.UnitarilyInvariantSeminorm.map_comp_starProjection_le {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 F] (N : UnitarilyInvariantSeminorm 𝕜 E F) (U : Submodule 𝕜 E) (A : E →L[𝕜] F) : N ↑(A ∘SL U.starProjection) ≤ N ↑A

Composing with an orthogonal projection on the right does not increase a unitarily invariant seminorm.

Ideal inequalities for a symmetric right factor bounded on a subspace

theorem TauCeti.UnitarilyInvariantSeminorm.map_comp_le_right_of_isSymmetric {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 F] (N : UnitarilyInvariantSeminorm 𝕜 E F) {A : E →ₗ[𝕜] F} {C : E →ₗ[𝕜] E} (hC : C.IsSymmetric) {T : Submodule 𝕜 E} (hA : ∀ x ∈ Tᗮ, A x = 0) {c : ℝ} (hCT : ∀ x ∈ T, ‖C x‖ ≤ c * ‖x‖) : N (A ∘ₗ C) ≤ c * N A

Ideal inequality, symmetric right factor bounded on a subspace. If A vanishes on Tᗮ, C is symmetric, and ‖C x‖ ≤ c * ‖x‖ for every x ∈ T, then N (A C) ≤ c * N A for every unitarily invariant seminorm N.

theorem TauCeti.UnitarilyInvariantSeminorm.mul_map_le_map_comp_right_of_isSymmetric {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 F] (N : UnitarilyInvariantSeminorm 𝕜 E F) {A : E →ₗ[𝕜] F} {C : E →ₗ[𝕜] E} (hC : C.IsSymmetric) {T : Submodule 𝕜 E} (hA : ∀ x ∈ Tᗮ, A x = 0) {c : ℝ} (hCT : ∀ x ∈ T, c * ‖x‖ ≤ ‖C x‖) : c * N A ≤ N (A ∘ₗ C)

Reverse ideal inequality, symmetric right factor bounded below on a subspace. If A vanishes on Tᗮ, C is symmetric, and c * ‖x‖ ≤ ‖C x‖ for every x ∈ T, then c * N A ≤ N (A C) for every unitarily invariant seminorm N.