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Mathlib.Analysis.InnerProductSpace.Symmetric

Symmetric linear maps in an inner product space #

This file defines and proves basic theorems about symmetric not necessarily bounded operators on an inner product space, i.e linear maps T : E → E such that ∀ x y, ⟪T x, y⟫ = ⟪x, T y⟫.

In comparison to IsSelfAdjoint, this definition works for non-continuous linear maps, and doesn't rely on the definition of the adjoint, which allows it to be stated in non-complete space.

Main definitions #

Main statements #

Tags #

self-adjoint, symmetric

Symmetric operators #

def LinearMap.IsSymmetric {𝕜 : Type u_1} {E : Type u_2} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (T : E →ₗ[𝕜] E) :

A (not necessarily bounded) operator on an inner product space is symmetric, if for all x, y, we have ⟪T x, y⟫ = ⟪x, T y⟫.

Equations
Instances For
    theorem LinearMap.isSymmetric_iff_sesqForm {𝕜 : Type u_1} {E : Type u_2} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (T : E →ₗ[𝕜] E) :

    An operator T on an inner product space is symmetric if and only if it is LinearMap.IsSelfAdjoint with respect to the sesquilinear form given by the inner product.

    theorem LinearMap.IsSymmetric.conj_inner_sym {𝕜 : Type u_1} {E : Type u_2} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : LinearMap.IsSymmetric T) (x : E) (y : E) :
    (starRingEnd 𝕜) T x, y⟫_𝕜 = T y, x⟫_𝕜
    @[simp]
    theorem LinearMap.IsSymmetric.apply_clm {𝕜 : Type u_1} {E : Type u_2} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →L[𝕜] E} (hT : LinearMap.IsSymmetric T) (x : E) (y : E) :
    T x, y⟫_𝕜 = x, T y⟫_𝕜
    theorem LinearMap.isSymmetric_id {𝕜 : Type u_1} {E : Type u_2} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] :
    theorem LinearMap.IsSymmetric.add {𝕜 : Type u_1} {E : Type u_2} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} {S : E →ₗ[𝕜] E} (hT : LinearMap.IsSymmetric T) (hS : LinearMap.IsSymmetric S) :
    theorem LinearMap.IsSymmetric.continuous {𝕜 : Type u_1} {E : Type u_2} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →ₗ[𝕜] E} (hT : LinearMap.IsSymmetric T) :

    The Hellinger--Toeplitz theorem: if a symmetric operator is defined on a complete space, then it is automatically continuous.

    @[simp]
    theorem LinearMap.IsSymmetric.coe_reApplyInnerSelf_apply {𝕜 : Type u_1} {E : Type u_2} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →L[𝕜] E} (hT : LinearMap.IsSymmetric T) (x : E) :
    (ContinuousLinearMap.reApplyInnerSelf T x) = T x, x⟫_𝕜

    For a symmetric operator T, the function fun x ↦ ⟪T x, x⟫ is real-valued.

    theorem LinearMap.IsSymmetric.restrict_invariant {𝕜 : Type u_1} {E : Type u_2} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : LinearMap.IsSymmetric T) {V : Submodule 𝕜 E} (hV : vV, T v V) :

    If a symmetric operator preserves a submodule, its restriction to that submodule is symmetric.

    A linear operator on a complex inner product space is symmetric precisely when ⟪T v, v⟫_ℂ is real for all v.

    theorem LinearMap.IsSymmetric.inner_map_polarization {𝕜 : Type u_1} {E : Type u_2} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : LinearMap.IsSymmetric T) (x : E) (y : E) :
    T x, y⟫_𝕜 = (T (x + y), x + y⟫_𝕜 - T (x - y), x - y⟫_𝕜 - IsROrC.I * T (x + IsROrC.I y), x + IsROrC.I y⟫_𝕜 + IsROrC.I * T (x - IsROrC.I y), x - IsROrC.I y⟫_𝕜) / 4

    Polarization identity for symmetric linear maps. See inner_map_polarization for the complex version without the symmetric assumption.

    theorem LinearMap.IsSymmetric.inner_map_self_eq_zero {𝕜 : Type u_1} {E : Type u_2} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : LinearMap.IsSymmetric T) :
    (∀ (x : E), T x, x⟫_𝕜 = 0) T = 0

    A symmetric linear map T is zero if and only if ⟪T x, x⟫_ℝ = 0 for all x. See inner_map_self_eq_zero for the complex version without the symmetric assumption.