Positive operators

In this file we define positive operators in a Hilbert space. We follow Bourbaki's choice of requiring self adjointness in the definition.

Main definitions

• IsPositive : a continuous linear map is positive if it is self adjoint and ∀ x, 0 ≤ re ⟪T x, x⟫

Main statements

• ContinuousLinearMap.IsPositive.conj_adjoint : if T : E →L[𝕜] E is positive, then for any S : E →L[𝕜] F, S ∘L T ∘L S† is also positive.
• ContinuousLinearMap.isPositive_iff_complex : in a complex Hilbert space, checking that ⟪T x, x⟫ is a nonnegative real number for all x suffices to prove that T is positive

References

• Bourbaki, Topological Vector Spaces

Tags

Positive operator

def ContinuousLinearMap.IsPositive {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (T : E →L[𝕜] E) : Prop

A continuous linear endomorphism T of a Hilbert space is positive if it is self adjoint and ∀ x, 0 ≤ re ⟪T x, x⟫.

Equations

• T.IsPositive = (IsSelfAdjoint T ∧ ∀ (x : E), 0 ≤ T.reApplyInnerSelf x)

theorem ContinuousLinearMap.IsPositive.isSelfAdjoint {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : T.IsPositive) : IsSelfAdjoint T

theorem ContinuousLinearMap.IsPositive.inner_nonneg_left {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : T.IsPositive) (x : E) : 0 ≤ RCLike.re (inner (T x) x)

theorem ContinuousLinearMap.IsPositive.inner_nonneg_right {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : T.IsPositive) (x : E) : 0 ≤ RCLike.re (inner x (T x))

theorem ContinuousLinearMap.isPositive_zero {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] : ContinuousLinearMap.IsPositive 0

theorem ContinuousLinearMap.isPositive_one {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] : ContinuousLinearMap.IsPositive 1

theorem ContinuousLinearMap.IsPositive.add {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPositive) : (T + S).IsPositive

theorem ContinuousLinearMap.IsPositive.conj_adjoint {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F) : (S.comp (T.comp (ContinuousLinearMap.adjoint S))).IsPositive

theorem ContinuousLinearMap.IsPositive.adjoint_conj {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) : ((ContinuousLinearMap.adjoint S).comp (T.comp S)).IsPositive

theorem ContinuousLinearMap.IsPositive.conj_orthogonalProjection {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (U : Submodule 𝕜 E) {T : E →L[𝕜] E} (hT : T.IsPositive) [CompleteSpace ↥U] : (U.subtypeL.comp ((orthogonalProjection U).comp (T.comp (U.subtypeL.comp (orthogonalProjection U))))).IsPositive

theorem ContinuousLinearMap.IsPositive.orthogonalProjection_comp {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E) [CompleteSpace ↥U] : ((orthogonalProjection U).comp (T.comp U.subtypeL)).IsPositive

theorem ContinuousLinearMap.antilipschitz_of_forall_le_inner_map {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_4} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] (f : H →L[𝕜] H) {c : NNReal} (hc : 0 < c) (h : ∀ (x : H), ‖x‖ ^ 2 * ↑c ≤ ‖inner (f x) x‖) : AntilipschitzWith c⁻¹ ⇑f

theorem ContinuousLinearMap.isUnit_of_forall_le_norm_inner_map {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (f : E →L[𝕜] E) {c : NNReal} (hc : 0 < c) (h : ∀ (x : E), ‖x‖ ^ 2 * ↑c ≤ ‖inner (f x) x‖) : IsUnit f

theorem ContinuousLinearMap.isPositive_iff_complex {E' : Type u_4} [NormedAddCommGroup E'] [InnerProductSpace ℂ E'] [CompleteSpace E'] (T : E' →L[ℂ] E') : T.IsPositive ↔ ∀ (x : E'), ↑(RCLike.re (inner (T x) x)) = inner (T x) x ∧ 0 ≤ RCLike.re (inner (T x) x)

instance ContinuousLinearMap.instLoewnerPartialOrder {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] : PartialOrder (E →L[𝕜] E)

The (Loewner) partial order on continuous linear maps on a Hilbert space determined by f ≤ g if and only if g - f is a positive linear map (in the sense of ContinuousLinearMap.IsPositive). With this partial order, the continuous linear maps form a StarOrderedRing.

Equations

• ContinuousLinearMap.instLoewnerPartialOrder = PartialOrder.mk ⋯

theorem ContinuousLinearMap.le_def {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (f g : E →L[𝕜] E) : f ≤ g ↔ (g - f).IsPositive

theorem ContinuousLinearMap.nonneg_iff_isPositive {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (f : E →L[𝕜] E) : 0 ≤ f ↔ f.IsPositive