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analysis.inner_product_space.symmetric

Symmetric linear maps in an inner product space #

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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 is_self_adjoint, 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 #

• linear_map.is_symmetric: 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⟫

Main statements #

• is_symmetric.continuous: if a symmetric operator is defined on a complete space, then it is automatically continuous.

Tags #

Symmetric operators #

def linear_map.is_symmetric {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] (T : E →ₗ[𝕜] E) :
Prop

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
theorem linear_map.is_symmetric_iff_sesq_form {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] (T : E →ₗ[𝕜] E) :

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

theorem linear_map.is_symmetric.conj_inner_sym {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] {T : E →ₗ[𝕜] E} (hT : T.is_symmetric) (x y : E) :
@[simp]
theorem linear_map.is_symmetric.apply_clm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] {T : E →L[𝕜] E} (hT : T.is_symmetric) (x y : E) :
has_inner.inner (T x) y = (T y)
theorem linear_map.is_symmetric_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] :
theorem linear_map.is_symmetric_id {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] :
theorem linear_map.is_symmetric.add {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] {T S : E →ₗ[𝕜] E} (hT : T.is_symmetric) (hS : S.is_symmetric) :
theorem linear_map.is_symmetric.continuous {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] {T : E →ₗ[𝕜] E} (hT : T.is_symmetric) :

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

@[simp]
theorem linear_map.is_symmetric.coe_re_apply_inner_self_apply {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] {T : E →L[𝕜] E} (hT : T.is_symmetric) (x : E) :

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

theorem linear_map.is_symmetric.restrict_invariant {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] {T : E →ₗ[𝕜] E} (hT : T.is_symmetric) {V : E} (hV : (v : E), v V T v V) :

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

theorem linear_map.is_symmetric.restrict_scalars {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] {T : E →ₗ[𝕜] E} (hT : T.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 linear_map.is_symmetric.inner_map_polarization {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] {T : E →ₗ[𝕜] E} (hT : T.is_symmetric) (x y : E) :

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

theorem linear_map.is_symmetric.inner_map_self_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] {T : E →ₗ[𝕜] E} (hT : T.is_symmetric) :
( (x : E), has_inner.inner (T x) x = 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.