The spectrum of elements in a complete normed algebra

This file contains the basic theory for the resolvent and spectrum of a Banach algebra.

Main definitions

• spectralRadius : ℝ≥0∞: supremum of ‖k‖₊ for all k ∈ spectrum 𝕜 a
• NormedRing.algEquivComplexOfComplete: Gelfand-Mazur theorem For a complex Banach division algebra, the natural algebraMap ℂ A is an algebra isomorphism whose inverse is given by selecting the (unique) element of spectrum ℂ a

Main statements

• spectrum.isOpen_resolventSet: the resolvent set is open.
• spectrum.isClosed: the spectrum is closed.
• spectrum.subset_closedBall_norm: the spectrum is a subset of closed disk of radius equal to the norm.
• spectrum.isCompact: the spectrum is compact.
• spectrum.spectralRadius_le_nnnorm: the spectral radius is bounded above by the norm.
• spectrum.hasDerivAt_resolvent: the resolvent function is differentiable on the resolvent set.
• spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius: Gelfand's formula for the spectral radius in Banach algebras over ℂ.
• spectrum.nonempty: the spectrum of any element in a complex Banach algebra is nonempty.

TODO

• compute all derivatives of resolvent a.

noncomputable def spectralRadius (𝕜 : Type u_1) {A : Type u_2} [NormedField 𝕜] [Ring A] [Algebra 𝕜 A] (a : A) : ENNReal

The spectral radius is the supremum of the nnnorm (‖·‖₊) of elements in the spectrum, coerced into an element of ℝ≥0∞. Note that it is possible for spectrum 𝕜 a = ∅. In this case, spectralRadius a = 0. It is also possible that spectrum 𝕜 a be unbounded (though not for Banach algebras, see spectrum.is_bounded, below). In this case, spectralRadius a = ∞.

@[simp]

theorem spectrum.SpectralRadius.of_subsingleton {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [Subsingleton A] (a : A) : spectralRadius 𝕜 a = 0

@[simp]

theorem spectrum.spectralRadius_zero {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] : spectralRadius 𝕜 0 = 0

theorem spectrum.mem_resolventSet_of_spectralRadius_lt {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] {a : A} {k : 𝕜} (h : spectralRadius 𝕜 a < ↑‖k‖₊) : k ∈ resolventSet 𝕜 a

theorem spectrum.isOpen_resolventSet {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) : IsOpen (resolventSet 𝕜 a)

theorem spectrum.isClosed {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) : IsClosed (spectrum 𝕜 a)

theorem spectrum.mem_resolventSet_of_norm_lt_mul {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] {a : A} {k : 𝕜} (h : ‖a‖ * ‖1‖ < ‖k‖) : k ∈ resolventSet 𝕜 a

theorem spectrum.mem_resolventSet_of_norm_lt {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [NormOneClass A] {a : A} {k : 𝕜} (h : ‖a‖ < ‖k‖) : k ∈ resolventSet 𝕜 a

theorem spectrum.norm_le_norm_mul_of_mem {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] {a : A} {k : 𝕜} (hk : k ∈ spectrum 𝕜 a) : ‖k‖ ≤ ‖a‖ * ‖1‖

theorem spectrum.norm_le_norm_of_mem {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [NormOneClass A] {a : A} {k : 𝕜} (hk : k ∈ spectrum 𝕜 a) : ‖k‖ ≤ ‖a‖

theorem spectrum.subset_closedBall_norm_mul {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) : spectrum 𝕜 a ⊆ Metric.closedBall 0 (‖a‖ * ‖1‖)

theorem spectrum.subset_closedBall_norm {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [NormOneClass A] (a : A) : spectrum 𝕜 a ⊆ Metric.closedBall 0 ‖a‖

theorem spectrum.isBounded {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) : Bornology.IsBounded (spectrum 𝕜 a)

theorem spectrum.isCompact {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [ProperSpace 𝕜] (a : A) : IsCompact (spectrum 𝕜 a)

theorem spectrum.spectralRadius_le_nnnorm {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [NormOneClass A] (a : A) : spectralRadius 𝕜 a ≤ ↑‖a‖₊

theorem spectrum.exists_nnnorm_eq_spectralRadius_of_nonempty {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [ProperSpace 𝕜] {a : A} (ha : Set.Nonempty (spectrum 𝕜 a)) : ∃ k, k ∈ spectrum 𝕜 a ∧ ↑‖k‖₊ = spectralRadius 𝕜 a

theorem spectrum.spectralRadius_lt_of_forall_lt_of_nonempty {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [ProperSpace 𝕜] {a : A} (ha : Set.Nonempty (spectrum 𝕜 a)) {r : NNReal} (hr : ∀ (k : 𝕜), k ∈ spectrum 𝕜 a → ‖k‖₊ < r) : spectralRadius 𝕜 a < ↑r

theorem spectrum.spectralRadius_le_pow_nnnorm_pow_one_div (𝕜 : Type u_1) {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) (n : ℕ) : spectralRadius 𝕜 a ≤ ↑‖a ^ (n + 1)‖₊ ^ (1 / (↑n + 1)) * ↑‖1‖₊ ^ (1 / (↑n + 1))

theorem spectrum.spectralRadius_le_liminf_pow_nnnorm_pow_one_div (𝕜 : Type u_1) {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) : spectralRadius 𝕜 a ≤ Filter.liminf (fun n => ↑‖a ^ n‖₊ ^ (1 / ↑n)) Filter.atTop

theorem spectrum.hasDerivAt_resolvent {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] {a : A} {k : 𝕜} (hk : k ∈ resolventSet 𝕜 a) : HasDerivAt (resolvent a) (-resolvent a k ^ 2) k

theorem spectrum.norm_resolvent_le_forall {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) (ε : ℝ) : ε > 0 → ∃ R, R > 0 ∧ ∀ (z : 𝕜), R ≤ ‖z‖ → ‖resolvent a z‖ ≤ ε

theorem spectrum.hasFPowerSeriesOnBall_inverse_one_sub_smul (𝕜 : Type u_1) {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) : HasFPowerSeriesOnBall (fun z => Ring.inverse (1 - z • a)) (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 (↑‖a‖₊)⁻¹

In a Banach algebra A over a nontrivially normed field 𝕜, for any a : A the power series with coefficients a ^ n represents the function (1 - z • a)⁻¹ in a disk of radius ‖a‖₊⁻¹.

theorem spectrum.isUnit_one_sub_smul_of_lt_inv_radius {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] {a : A} {z : 𝕜} (h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹) : IsUnit (1 - z • a)

theorem spectrum.differentiableOn_inverse_one_sub_smul {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] {a : A} {r : NNReal} (hr : ↑r < (spectralRadius 𝕜 a)⁻¹) : DifferentiableOn 𝕜 (fun z => Ring.inverse (1 - z • a)) (Metric.closedBall 0 ↑r)

In a Banach algebra A over 𝕜, for a : A the function fun z ↦ (1 - z • a)⁻¹ is differentiable on any closed ball centered at zero of radius r < (spectralRadius 𝕜 a)⁻¹.

theorem spectrum.limsup_pow_nnnorm_pow_one_div_le_spectralRadius {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] (a : A) : Filter.limsup (fun n => ↑‖a ^ n‖₊ ^ (1 / ↑n)) Filter.atTop ≤ spectralRadius ℂ a

The limsup relationship for the spectral radius used to prove spectrum.gelfand_formula.

theorem spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] (a : A) : Filter.Tendsto (fun n => ↑‖a ^ n‖₊ ^ (1 / ↑n)) Filter.atTop (nhds (spectralRadius ℂ a))

Gelfand's formula: Given an element a : A of a complex Banach algebra, the spectralRadius of a is the limit of the sequence ‖a ^ n‖₊ ^ (1 / n).

theorem spectrum.pow_norm_pow_one_div_tendsto_nhds_spectralRadius {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] (a : A) : Filter.Tendsto (fun n => ENNReal.ofReal (‖a ^ n‖ ^ (1 / ↑n))) Filter.atTop (nhds (spectralRadius ℂ a))

Gelfand's formula: Given an element a : A of a complex Banach algebra, the spectralRadius of a is the limit of the sequence ‖a ^ n‖₊ ^ (1 / n).

theorem spectrum.nonempty {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [Nontrivial A] (a : A) : Set.Nonempty (spectrum ℂ a)

In a (nontrivial) complex Banach algebra, every element has nonempty spectrum.

theorem spectrum.exists_nnnorm_eq_spectralRadius {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [Nontrivial A] (a : A) : ∃ z, z ∈ spectrum ℂ a ∧ ↑‖z‖₊ = spectralRadius ℂ a

In a complex Banach algebra, the spectral radius is always attained by some element of the spectrum.

theorem spectrum.spectralRadius_lt_of_forall_lt {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [Nontrivial A] (a : A) {r : NNReal} (hr : ∀ (z : ℂ), z ∈ spectrum ℂ a → ‖z‖₊ < r) : spectralRadius ℂ a < ↑r

In a complex Banach algebra, if every element of the spectrum has norm strictly less than r : ℝ≥0, then the spectral radius is also strictly less than r.

theorem spectrum.map_polynomial_aeval {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [Nontrivial A] (a : A) (p : Polynomial ℂ) : spectrum ℂ (↑(Polynomial.aeval a) p) = (fun k => Polynomial.eval k p) '' spectrum ℂ a

The spectral mapping theorem for polynomials in a Banach algebra over ℂ.

theorem spectrum.map_pow {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [Nontrivial A] (a : A) (n : ℕ) : spectrum ℂ (a ^ n) = (fun x => x ^ n) '' spectrum ℂ a

A specialization of the spectral mapping theorem for polynomials in a Banach algebra over ℂ to monic monomials.

theorem spectrum.algebraMap_eq_of_mem {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] (hA : ∀ {a : A}, IsUnit a ↔ a ≠ 0) {a : A} {z : ℂ} (h : z ∈ spectrum ℂ a) : ↑(algebraMap ℂ A) z = a

@[simp]

theorem NormedRing.algEquivComplexOfComplete_symm_apply {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] (hA : ∀ {a : A}, IsUnit a ↔ a ≠ 0) [CompleteSpace A] (a : A) : ↑(AlgEquiv.symm (NormedRing.algEquivComplexOfComplete hA)) a = Set.Nonempty.some (_ : Set.Nonempty (spectrum ℂ a))

@[simp]

theorem NormedRing.algEquivComplexOfComplete_apply {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] (hA : ∀ {a : A}, IsUnit a ↔ a ≠ 0) [CompleteSpace A] (a : ℂ) : ↑(NormedRing.algEquivComplexOfComplete hA) a = ↑(algebraMap ℂ A) a

noncomputable def NormedRing.algEquivComplexOfComplete {A : Type u_2} [NormedRing A] [NormedAlgebra ℂ A] (hA : ∀ {a : A}, IsUnit a ↔ a ≠ 0) [CompleteSpace A] : ℂ ≃ₐ[ℂ] A

Gelfand-Mazur theorem: For a complex Banach division algebra, the natural algebraMap ℂ A is an algebra isomorphism whose inverse is given by selecting the (unique) element of spectrum ℂ a. In addition, algebraMap_isometry guarantees this map is an isometry.

Note: because NormedDivisionRing requires the field norm_mul' : ∀ a b, ‖a * b‖ = ‖a‖ * ‖b‖, we don't use this type class and instead opt for a NormedRing in which the nonzero elements are precisely the units. This allows for the application of this isomorphism in broader contexts, e.g., to the quotient of a complex Banach algebra by a maximal ideal. In the case when A is actually a NormedDivisionRing, one may fill in the argument hA with the lemma isUnit_iff_ne_zero.

theorem spectrum.exp_mem_exp {𝕜 : Type u_1} {A : Type u_2} [IsROrC 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) {z : 𝕜} (hz : z ∈ spectrum 𝕜 a) : exp 𝕜 z ∈ spectrum 𝕜 (exp 𝕜 a)

For 𝕜 = ℝ or 𝕜 = ℂ, exp 𝕜 maps the spectrum of a into the spectrum of exp 𝕜 a.

instance AlgHom.instContinuousLinearMapClassToSemiringToDivisionSemiringToSemifieldToFieldToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingToRingToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingToRingToNormedRingToModuleToSeminormedAddCommGroupToNonUnitalSeminormedRingToNonUnitalNormedRingToNormedSpace'ToModuleToSeminormedAddCommGroupToNonUnitalSeminormedRingToNonUnitalNormedRingToNormedSpace {𝕜 : Type u_1} {A : Type u_2} {F : Type u_3} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [AlgHomClass F 𝕜 A 𝕜] : ContinuousLinearMapClass F 𝕜 A 𝕜

An algebra homomorphism into the base field, as a continuous linear map (since it is automatically bounded). See note [lower instance priority]

def AlgHom.toContinuousLinearMap {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (φ : A →ₐ[𝕜] 𝕜) : A →L[𝕜] 𝕜

An algebra homomorphism into the base field, as a continuous linear map (since it is automatically bounded).

@[simp]

theorem AlgHom.coe_toContinuousLinearMap {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (φ : A →ₐ[𝕜] 𝕜) : ↑(AlgHom.toContinuousLinearMap φ) = ↑φ

theorem AlgHom.norm_apply_le_self_mul_norm_one {𝕜 : Type u_1} {A : Type u_2} {F : Type u_3} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [AlgHomClass F 𝕜 A 𝕜] (f : F) (a : A) : ‖↑f a‖ ≤ ‖a‖ * ‖1‖

theorem AlgHom.norm_apply_le_self {𝕜 : Type u_1} {A : Type u_2} {F : Type u_3} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [NormOneClass A] [AlgHomClass F 𝕜 A 𝕜] (f : F) (a : A) : ‖↑f a‖ ≤ ‖a‖

@[simp]

theorem AlgHom.toContinuousLinearMap_norm {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [NormOneClass A] (φ : A →ₐ[𝕜] 𝕜) : ‖AlgHom.toContinuousLinearMap φ‖ = 1

def WeakDual.CharacterSpace.equivAlgHom {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [CompleteSpace A] [NormedAlgebra 𝕜 A] : ↑(WeakDual.characterSpace 𝕜 A) ≃ (A →ₐ[𝕜] 𝕜)

The equivalence between characters and algebra homomorphisms into the base field.

@[simp]

theorem WeakDual.CharacterSpace.equivAlgHom_coe {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [CompleteSpace A] [NormedAlgebra 𝕜 A] (f : ↑(WeakDual.characterSpace 𝕜 A)) : ↑(↑WeakDual.CharacterSpace.equivAlgHom f) = ↑f

@[simp]

theorem WeakDual.CharacterSpace.equivAlgHom_symm_coe {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [CompleteSpace A] [NormedAlgebra 𝕜 A] (f : A →ₐ[𝕜] 𝕜) : ↑(↑WeakDual.CharacterSpace.equivAlgHom.symm f) = ↑f