The spectrum of elements in a complete normed algebra #

This file contains the basic theory for the resolvent and spectrum of a Banach algebra. Theorems specific to complex Banach algebras, such as Gelfand's formula can be found in Mathlib/Analysis/Normed/Algebra/GelfandFormula.lean.

Main definitions #

spectralRadius : ℝ≥0∞: supremum of ‖k‖₊ for all k ∈ spectrum 𝕜 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.

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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.isBounded, below). In this case, spectralRadius a = ∞.

Equations

spectralRadius 𝕜 a = ⨆ k ∈ spectrum 𝕜 a, ↑‖k‖₊

Instances For

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@[simp]

theorem spectrum.SpectralRadius.of_subsingleton {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [Ring A] [Algebra 𝕜 A] [Subsingleton A] (a : A) :

spectralRadius 𝕜 a = 0

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@[simp]

theorem spectrum.spectralRadius_zero {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [Ring A] [Algebra 𝕜 A] :

spectralRadius 𝕜 0 = 0

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@[simp]

theorem spectrum.spectralRadius_one {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [Ring A] [Algebra 𝕜 A] [Nontrivial A] :

spectralRadius 𝕜 1 = 1

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

k ∈ resolventSet 𝕜 a

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theorem spectrum.spectralRadius_pow_le {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [Ring A] [Algebra 𝕜 A] (a : A) (n : ℕ) (hn : n ≠ 0) :

spectralRadius 𝕜 a ^ n ≤ spectralRadius 𝕜 (a ^ n)

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theorem spectrum.spectralRadius_pow_le' {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [Ring A] [Algebra 𝕜 A] [Nontrivial A] (a : A) (n : ℕ) :

spectralRadius 𝕜 a ^ n ≤ spectralRadius 𝕜 (a ^ n)

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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)

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@[simp]

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)

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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

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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

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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‖

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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‖

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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‖)

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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‖

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@[simp]

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)

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@[simp]

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)

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

CompactSpace ↑(spectrum 𝕜 a)

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instance spectrum.instCompactSpaceNNReal {A : Type u_3} [NormedRing A] [NormedAlgebra ℝ A] (a : A) [CompactSpace ↑(spectrum ℝ a)] :

CompactSpace ↑(spectrum NNReal a)

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@[simp]

theorem spectrum.isCompact_nnreal {A : Type u_3} [NormedRing A] [NormedAlgebra ℝ A] (a : A) [CompactSpace ↑(spectrum ℝ a)] :

IsCompact (spectrum NNReal a)

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@[simp]

theorem quasispectrum.isCompact {𝕜 : Type u_1} [NormedField 𝕜] {B : Type u_3} [NonUnitalNormedRing B] [NormedSpace 𝕜 B] [CompleteSpace B] [IsScalarTower 𝕜 B B] [SMulCommClass 𝕜 B B] [ProperSpace 𝕜] (a : B) :

IsCompact (quasispectrum 𝕜 a)

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instance quasispectrum.instCompactSpace {𝕜 : Type u_1} [NormedField 𝕜] {B : Type u_3} [NonUnitalNormedRing B] [NormedSpace 𝕜 B] [CompleteSpace B] [IsScalarTower 𝕜 B B] [SMulCommClass 𝕜 B B] [ProperSpace 𝕜] (a : B) :

CompactSpace ↑(quasispectrum 𝕜 a)

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instance quasispectrum.instCompactSpaceNNReal {B : Type u_3} [NonUnitalNormedRing B] [NormedSpace ℝ B] [IsScalarTower ℝ B B] [SMulCommClass ℝ B B] (a : B) [CompactSpace ↑(quasispectrum ℝ a)] :

CompactSpace ↑(quasispectrum NNReal a)

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@[simp]

theorem quasispectrum.isCompact_nnreal {B : Type u_3} [NonUnitalNormedRing B] [NormedSpace ℝ B] [IsScalarTower ℝ B B] [SMulCommClass ℝ B B] (a : B) [CompactSpace ↑(quasispectrum ℝ a)] :

IsCompact (quasispectrum NNReal a)

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theorem spectrum.le_nnnorm_of_mem {A : Type u_3} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] [NormOneClass A] {a : A} {r : NNReal} (hr : r ∈ spectrum NNReal a) :

r ≤ ‖a‖₊

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theorem spectrum.coe_le_norm_of_mem {A : Type u_3} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] [NormOneClass A] {a : A} {r : NNReal} (hr : r ∈ spectrum NNReal a) :

↑r ≤ ‖a‖

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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‖₊

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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 : (spectrum 𝕜 a).Nonempty) :

∃ k ∈ spectrum 𝕜 a, ↑‖k‖₊ = spectralRadius 𝕜 a

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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 : (spectrum 𝕜 a).Nonempty) {r : NNReal} (hr : ∀ k ∈ spectrum 𝕜 a, ‖k‖₊ < r) :

spectralRadius 𝕜 a < ↑r

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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))

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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 : ℕ) => ↑‖a ^ n‖₊ ^ (1 / ↑n)) Filter.atTop

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theorem spectrum.eventually_isUnit_resolvent {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) :

∀ᶠ (z : 𝕜) in Bornology.cobounded 𝕜, IsUnit (resolvent a z)

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theorem spectrum.resolvent_isBigO_inv {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) :

resolvent a =O[Bornology.cobounded 𝕜] Inv.inv

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theorem spectrum.resolvent_tendsto_cobounded {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A) :

Filter.Tendsto (resolvent a) (Bornology.cobounded 𝕜) (nhds 0)

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theorem spectrum.hasFPowerSeriesOnBall_inverse_one_sub_smul (𝕜 : Type u_1) {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [HasSummableGeomSeries A] (a : A) :

HasFPowerSeriesOnBall (fun (z : 𝕜) => Ring.inverse (1 - z • a)) (fun (n : ℕ) => ContinuousMultilinearMap.mkPiRing 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.

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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)

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

NormedSpace.exp 𝕜 z ∈ spectrum 𝕜 (NormedSpace.exp 𝕜 a)

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

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@[instance 100]

instance AlgHom.instContinuousLinearMapClassOfAlgHomClass {𝕜 : Type u_1} {A : Type u_2} {F : Type u_3} [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] [FunLike F A 𝕜] [AlgHomClass F 𝕜 A 𝕜] :

ContinuousLinearMapClass F 𝕜 A 𝕜

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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) :

StrongDual 𝕜 A

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

Equations

φ.toContinuousLinearMap = { toLinearMap := φ.toLinearMap, cont := ⋯ }

Instances For

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@[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) :

⇑φ.toContinuousLinearMap = ⇑φ

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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] [FunLike F A 𝕜] [AlgHomClass F 𝕜 A 𝕜] (f : F) (a : A) :

‖f a‖ ≤ ‖a‖ * ‖1‖

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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] [FunLike F A 𝕜] [AlgHomClass F 𝕜 A 𝕜] (f : F) (a : A) :

‖f a‖ ≤ ‖a‖

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@[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) :

‖φ.toContinuousLinearMap‖ = 1

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def WeakDual.CharacterSpace.equivAlgHom {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [CompleteSpace A] [NormedAlgebra 𝕜 A] :

↑(characterSpace 𝕜 A) ≃ (A →ₐ[𝕜] 𝕜)

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

Equations

WeakDual.CharacterSpace.equivAlgHom = { toFun := WeakDual.CharacterSpace.toAlgHom, invFun := fun (f : A →ₐ[𝕜] 𝕜) => ⟨f.toContinuousLinearMap, ⋯⟩, left_inv := ⋯, right_inv := ⋯ }

Instances For

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@[simp]

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

⇑(equivAlgHom f) = ⇑f

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@[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) :

⇑(equivAlgHom.symm f) = ⇑f

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theorem Subalgebra.isUnit_of_isUnit_val_of_eventually {𝕜 : Type u_3} {A : Type u_4} {SA : Type u_5} [NormedRing A] [CompleteSpace A] [SetLike SA A] [SubringClass SA A] [NormedField 𝕜] [NormedAlgebra 𝕜 A] [instSMulMem : SMulMemClass SA 𝕜 A] (S : SA) [hS : IsClosed ↑S] {l : Filter ↥S} {a : ↥S} (ha : IsUnit ↑a) (hla : l ≤ nhds a) (hl : ∀ᶠ (x : ↥S) in l, IsUnit x) (hl' : l.NeBot) :

IsUnit a

Let S be a closed subalgebra of a Banach algebra A. If a : S is invertible in A, and for all x : S sufficiently close to a within some filter l, x is invertible in S, then a is invertible in S as well.

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theorem Subalgebra.frontier_spectrum {𝕜 : Type u_3} {A : Type u_4} {SA : Type u_5} [NormedRing A] [CompleteSpace A] [SetLike SA A] [SubringClass SA A] [NormedField 𝕜] [NormedAlgebra 𝕜 A] [instSMulMem : SMulMemClass SA 𝕜 A] (S : SA) [hS : IsClosed ↑S] (x : ↥S) :

frontier (spectrum 𝕜 x) ⊆ spectrum 𝕜 ↑x

If S : Subalgebra 𝕜 A is a closed subalgebra of a Banach algebra A, then for any x : S, the boundary of the spectrum of x relative to S is a subset of the spectrum of ↑x : A relative to A.

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theorem Subalgebra.frontier_subset_frontier {𝕜 : Type u_3} {A : Type u_4} {SA : Type u_5} [NormedRing A] [CompleteSpace A] [SetLike SA A] [SubringClass SA A] [NormedField 𝕜] [NormedAlgebra 𝕜 A] [instSMulMem : SMulMemClass SA 𝕜 A] (S : SA) [hS : IsClosed ↑S] (x : ↥S) :

frontier (spectrum 𝕜 x) ⊆ frontier (spectrum 𝕜 ↑x)

If S is a closed subalgebra of a Banach algebra A, then for any x : S, the boundary of the spectrum of x relative to S is a subset of the boundary of the spectrum of ↑x : A relative to A.

source

theorem Subalgebra.spectrum_sUnion_connectedComponentIn {𝕜 : Type u_3} {A : Type u_4} {SA : Type u_5} [NormedRing A] [CompleteSpace A] [SetLike SA A] [SubringClass SA A] [NormedField 𝕜] [NormedAlgebra 𝕜 A] [instSMulMem : SMulMemClass SA 𝕜 A] (S : SA) [hS : IsClosed ↑S] (x : ↥S) :

spectrum 𝕜 x = spectrum 𝕜 ↑x ∪ ⋃ z ∈ spectrum 𝕜 x \ spectrum 𝕜 ↑x, connectedComponentIn (spectrum 𝕜 ↑x)ᶜ z

If S is a closed subalgebra of a Banach algebra A, then for any x : S, the spectrum of x is the spectrum of ↑x : A along with the connected components of the complement of the spectrum of ↑x : A which contain an element of the spectrum of x : S.

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theorem Subalgebra.spectrum_isBounded_connectedComponentIn {𝕜 : Type u_3} {A : Type u_4} {SA : Type u_5} [NormedRing A] [CompleteSpace A] [SetLike SA A] [SubringClass SA A] [NormedField 𝕜] [NormedAlgebra 𝕜 A] [instSMulMem : SMulMemClass SA 𝕜 A] (S : SA) [hS : IsClosed ↑S] (x : ↥S) {z : 𝕜} (hz : z ∈ spectrum 𝕜 x) :

Bornology.IsBounded (connectedComponentIn (spectrum 𝕜 ↑x)ᶜ z)

Let S be a closed subalgebra of a Banach algebra A, and let x : S. If z is in the spectrum of x, then the connected component of z in the complement of the spectrum of ↑x : A is bounded (or else z actually belongs to the spectrum of ↑x : A).

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theorem Subalgebra.spectrum_eq_of_isPreconnected_compl {𝕜 : Type u_3} {A : Type u_4} {SA : Type u_5} [NormedRing A] [CompleteSpace A] [SetLike SA A] [SubringClass SA A] [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 A] [SMulMemClass SA 𝕜 A] (S : SA) [hS : IsClosed ↑S] (x : ↥S) (h : IsPreconnected (spectrum 𝕜 ↑x)ᶜ) :

spectrum 𝕜 x = spectrum 𝕜 ↑x

Let S be a closed subalgebra of a Banach algebra A. If for x : S the complement of the spectrum of ↑x : A is connected, then spectrum 𝕜 x = spectrum 𝕜 (x : A).

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theorem SpectrumRestricts.spectralRadius_eq {𝕜₁ : Type u_3} {𝕜₂ : Type u_4} {A : Type u_5} [NormedField 𝕜₁] [NormedField 𝕜₂] [NormedRing A] [NormedAlgebra 𝕜₁ A] [NormedAlgebra 𝕜₂ A] [NormedAlgebra 𝕜₁ 𝕜₂] [IsScalarTower 𝕜₁ 𝕜₂ A] {f : 𝕜₂ → 𝕜₁} {a : A} (h : SpectrumRestricts a f) :

spectralRadius 𝕜₁ a = spectralRadius 𝕜₂ a

If 𝕜₁ is a normed field contained as subfield of a larger normed field 𝕜₂, and if a : A is an element whose 𝕜₂ spectrum restricts to 𝕜₁, then the spectral radii over each scalar field coincide.

source