Spectrum of an element in an algebra

This file develops the basic theory of the spectrum of an element of an algebra. This theory will serve as the foundation for spectral theory in Banach algebras.

Main definitions

• resolventSet a : Set R: the resolvent set of an element a : A where A is an R-algebra.
• spectrum a : Set R: the spectrum of an element a : A where A is an R-algebra.
• resolvent : R → A: the resolvent function is fun r ↦ Ring.inverse (↑ₐ r - a), and hence when r ∈ resolvent R A, it is actually the inverse of the unit (↑ₐ r - a).

Main statements

• spectrum.unit_smul_eq_smul and spectrum.smul_eq_smul: units in the scalar ring commute (multiplication) with the spectrum, and over a field even 0 commutes with the spectrum.
• spectrum.left_add_coset_eq: elements of the scalar ring commute (addition) with the spectrum.
• spectrum.unit_mem_mul_comm and spectrum.preimage_units_mul_comm: the units (of R) in σ (a*b) coincide with those in σ (b*a).
• spectrum.scalar_eq: in a nontrivial algebra over a field, the spectrum of a scalar is a singleton.

Notation

• σ a : spectrum R a of a : A

def resolventSet (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] (a : A) : Set R

Given a commutative ring R and an R-algebra A, the resolvent set of a : A is the Set R consisting of those r : R for which r•1 - a is a unit of the algebra A.

Equations

• resolventSet R a = {r : R | IsUnit ((algebraMap R A) r - a)}

def spectrum (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] (a : A) : Set R

Given a commutative ring R and an R-algebra A, the spectrum of a : A is the Set R consisting of those r : R for which r•1 - a is not a unit of the algebra A.

The spectrum is simply the complement of the resolvent set.

Equations

• spectrum R a = (resolventSet R a)ᶜ

noncomputable def resolvent {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] (a : A) (r : R) : A

Given an a : A where A is an R-algebra, the resolvent is a map R → A which sends r : R to (algebraMap R A r - a)⁻¹ when r ∈ resolvent R A and 0 when r ∈ spectrum R A.

Equations

• resolvent a r = Ring.inverse ((algebraMap R A) r - a)

noncomputable def IsUnit.subInvSMul {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : Rˣ} {s : R} {a : A} (h : IsUnit (r • (algebraMap R A) s - a)) : Aˣ

The unit 1 - r⁻¹ • a constructed from r • 1 - a when the latter is a unit.

Equations

• h.subInvSMul = { val := (algebraMap R A) s - r⁻¹ • a, inv := r • ↑h.unit⁻¹, val_inv := ⋯, inv_val := ⋯ }

@[simp]

theorem IsUnit.val_subInvSMul {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : Rˣ} {s : R} {a : A} (h : IsUnit (r • (algebraMap R A) s - a)) : ↑h.subInvSMul = (algebraMap R A) s - r⁻¹ • a

@[simp]

theorem IsUnit.val_inv_subInvSMul {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : Rˣ} {s : R} {a : A} (h : IsUnit (r • (algebraMap R A) s - a)) : ↑h.subInvSMul⁻¹ = r • ↑h.unit⁻¹

theorem spectrum.mem_iff {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : R} {a : A} : r ∈ spectrum R a ↔ ¬IsUnit ((algebraMap R A) r - a)

theorem spectrum.notMem_iff {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : R} {a : A} : r ∉ spectrum R a ↔ IsUnit ((algebraMap R A) r - a)

@[deprecated spectrum.notMem_iff (since := "2025-05-23")]

theorem spectrum.not_mem_iff {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : R} {a : A} : r ∉ spectrum R a ↔ IsUnit ((algebraMap R A) r - a)

Alias of spectrum.notMem_iff.

theorem spectrum.zero_mem_iff (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} : 0 ∈ spectrum R a ↔ ¬IsUnit a

theorem spectrum.zero_mem (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} : ¬IsUnit a → 0 ∈ spectrum R a

Alias of the reverse direction of spectrum.zero_mem_iff.

theorem spectrum.not_isUnit_of_zero_mem (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} : 0 ∈ spectrum R a → ¬IsUnit a

Alias of the forward direction of spectrum.zero_mem_iff.

theorem spectrum.zero_notMem_iff (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} : 0 ∉ spectrum R a ↔ IsUnit a

@[deprecated spectrum.zero_notMem_iff (since := "2025-05-23")]

theorem spectrum.zero_not_mem_iff (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} : 0 ∉ spectrum R a ↔ IsUnit a

Alias of spectrum.zero_notMem_iff.

theorem spectrum.zero_notMem (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} : IsUnit a → 0 ∉ spectrum R a

Alias of the reverse direction of spectrum.zero_notMem_iff.

---

Alias of spectrum.zero_notMem_iff.

theorem spectrum.isUnit_of_zero_notMem (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} : 0 ∉ spectrum R a → IsUnit a

Alias of the forward direction of spectrum.zero_notMem_iff.

---

Alias of spectrum.zero_notMem_iff.

@[deprecated spectrum.isUnit_of_zero_notMem (since := "2025-05-23")]

theorem spectrum.isUnit_of_zero_not_mem (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} : 0 ∉ spectrum R a → IsUnit a

Alias of the forward direction of spectrum.zero_notMem_iff.

---

Alias of the forward direction of spectrum.zero_notMem_iff.

---

Alias of spectrum.zero_notMem_iff.

@[deprecated spectrum.zero_notMem (since := "2025-05-23")]

theorem spectrum.zero_not_mem (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} : IsUnit a → 0 ∉ spectrum R a

Alias of the reverse direction of spectrum.zero_notMem_iff.

---

Alias of the reverse direction of spectrum.zero_notMem_iff.

---

Alias of spectrum.zero_notMem_iff.

@[simp]

theorem Units.zero_notMem_spectrum (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] (a : Aˣ) : 0 ∉ spectrum R ↑a

@[deprecated Units.zero_notMem_spectrum (since := "2025-05-23")]

theorem Units.zero_not_mem_spectrum (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] (a : Aˣ) : 0 ∉ spectrum R ↑a

Alias of Units.zero_notMem_spectrum.

theorem spectrum.subset_singleton_zero_compl (R : Type u) {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} (ha : IsUnit a) : spectrum R a ⊆ {0}ᶜ

theorem spectrum.mem_resolventSet_of_left_right_inverse {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : R} {a b c : A} (h₁ : ((algebraMap R A) r - a) * b = 1) (h₂ : c * ((algebraMap R A) r - a) = 1) : r ∈ resolventSet R a

theorem spectrum.mem_resolventSet_iff {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : R} {a : A} : r ∈ resolventSet R a ↔ IsUnit ((algebraMap R A) r - a)

@[simp]

theorem spectrum.algebraMap_mem_iff (S : Type u_1) {R : Type u_2} {A : Type u_3} [CommSemiring R] [CommSemiring S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] {a : A} {r : R} : (algebraMap R S) r ∈ spectrum S a ↔ r ∈ spectrum R a

theorem spectrum.of_algebraMap_mem (S : Type u_1) {R : Type u_2} {A : Type u_3} [CommSemiring R] [CommSemiring S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] {a : A} {r : R} : (algebraMap R S) r ∈ spectrum S a → r ∈ spectrum R a

Alias of the forward direction of spectrum.algebraMap_mem_iff.

theorem spectrum.algebraMap_mem (S : Type u_1) {R : Type u_2} {A : Type u_3} [CommSemiring R] [CommSemiring S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] {a : A} {r : R} : r ∈ spectrum R a → (algebraMap R S) r ∈ spectrum S a

Alias of the reverse direction of spectrum.algebraMap_mem_iff.

@[simp]

theorem spectrum.preimage_algebraMap (S : Type u_1) {R : Type u_2} {A : Type u_3} [CommSemiring R] [CommSemiring S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] {a : A} : ⇑(algebraMap R S) ⁻¹' spectrum S a = spectrum R a

@[simp]

theorem spectrum.resolventSet_of_subsingleton {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] [Subsingleton A] (a : A) : resolventSet R a = Set.univ

@[simp]

theorem spectrum.of_subsingleton {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] [Subsingleton A] (a : A) : spectrum R a = ∅

theorem spectrum.resolvent_eq {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} {r : R} (h : r ∈ resolventSet R a) : resolvent a r = ↑(IsUnit.unit h)⁻¹

theorem spectrum.units_smul_resolvent {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : Rˣ} {s : R} {a : A} : r • resolvent a s = resolvent (r⁻¹ • a) (r⁻¹ • s)

theorem spectrum.units_smul_resolvent_self {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : Rˣ} {a : A} : r • resolvent a ↑r = resolvent (r⁻¹ • a) 1

theorem spectrum.isUnit_resolvent {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : R} {a : A} : r ∈ resolventSet R a ↔ IsUnit (resolvent a r)

The resolvent is a unit when the argument is in the resolvent set.

theorem spectrum.inv_mem_resolventSet {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : Rˣ} {a : Aˣ} (h : ↑r ∈ resolventSet R ↑a) : ↑r⁻¹ ∈ resolventSet R ↑a⁻¹

theorem spectrum.inv_mem_iff {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {r : Rˣ} {a : Aˣ} : ↑r ∈ spectrum R ↑a ↔ ↑r⁻¹ ∈ spectrum R ↑a⁻¹

theorem spectrum.zero_mem_resolventSet_of_unit {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] (a : Aˣ) : 0 ∈ resolventSet R ↑a

theorem spectrum.ne_zero_of_mem_of_unit {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : Aˣ} {r : R} (hr : r ∈ spectrum R ↑a) : r ≠ 0

theorem spectrum.add_mem_iff {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} {r s : R} : r + s ∈ spectrum R a ↔ r ∈ spectrum R (-(algebraMap R A) s + a)

theorem spectrum.add_mem_add_iff {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} {r s : R} : r + s ∈ spectrum R ((algebraMap R A) s + a) ↔ r ∈ spectrum R a

theorem spectrum.smul_mem_smul_iff {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a : A} {s : R} {r : Rˣ} : r • s ∈ spectrum R (r • a) ↔ s ∈ spectrum R a

theorem spectrum.unit_smul_eq_smul {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] (a : A) (r : Rˣ) : spectrum R (r • a) = r • spectrum R a

theorem spectrum.unit_mem_mul_comm {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] {a b : A} {r : Rˣ} : ↑r ∈ spectrum R (a * b) ↔ ↑r ∈ spectrum R (b * a)

theorem spectrum.preimage_units_mul_comm {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] (a b : A) : Units.val ⁻¹' spectrum R (a * b) = Units.val ⁻¹' spectrum R (b * a)

theorem spectrum.setOf_isUnit_inter_mul_comm {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] (a b : A) : {r : R | IsUnit r} ∩ spectrum R (a * b) = {r : R | IsUnit r} ∩ spectrum R (b * a)

theorem spectrum.star_mem_resolventSet_iff {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] [InvolutiveStar R] [StarRing A] [StarModule R A] {r : R} {a : A} : star r ∈ resolventSet R a ↔ r ∈ resolventSet R (star a)

theorem spectrum.map_star {R : Type u} {A : Type v} [CommSemiring R] [Ring A] [Algebra R A] [InvolutiveStar R] [StarRing A] [StarModule R A] (a : A) : spectrum R (star a) = star (spectrum R a)

theorem spectrum.subset_subalgebra {S : Type u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Ring A] [Algebra R A] [SetLike S A] [SubringClass S A] [SMulMemClass S R A] {s : S} (a : ↥s) : spectrum R ↑a ⊆ spectrum R a

theorem spectrum.singleton_add_eq {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] (a : A) (r : R) : {r} + spectrum R a = spectrum R ((algebraMap R A) r + a)

theorem spectrum.add_singleton_eq {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] (a : A) (r : R) : spectrum R a + {r} = spectrum R (a + (algebraMap R A) r)

theorem spectrum.vadd_eq {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] (a : A) (r : R) : r +ᵥ spectrum R a = spectrum R ((algebraMap R A) r + a)

theorem spectrum.neg_eq {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] (a : A) : -spectrum R a = spectrum R (-a)

theorem spectrum.singleton_sub_eq {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] (a : A) (r : R) : {r} - spectrum R a = spectrum R ((algebraMap R A) r - a)

theorem spectrum.sub_singleton_eq {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] (a : A) (r : R) : spectrum R a - {r} = spectrum R (a - (algebraMap R A) r)

@[simp]

theorem spectrum.inv₀_mem_iff {R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A] {r : R} {a : Aˣ} : r⁻¹ ∈ spectrum R ↑a ↔ r ∈ spectrum R ↑a⁻¹

theorem spectrum.inv₀_mem_inv_iff {R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A] {r : R} {a : Aˣ} : r⁻¹ ∈ spectrum R ↑a⁻¹ ↔ r ∈ spectrum R ↑a

theorem spectrum.of_inv₀_mem {R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A] {r : R} {a : Aˣ} : r⁻¹ ∈ spectrum R ↑a → r ∈ spectrum R ↑a⁻¹

Alias of the forward direction of spectrum.inv₀_mem_iff.

theorem spectrum.inv₀_mem {R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A] {r : R} {a : Aˣ} : r ∈ spectrum R ↑a⁻¹ → r⁻¹ ∈ spectrum R ↑a

Alias of the reverse direction of spectrum.inv₀_mem_iff.

theorem spectrum.inv₀_mem_inv {R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A] {r : R} {a : Aˣ} : r ∈ spectrum R ↑a → r⁻¹ ∈ spectrum R ↑a⁻¹

Alias of the reverse direction of spectrum.inv₀_mem_inv_iff.

theorem spectrum.of_inv₀_mem_inv {R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A] {r : R} {a : Aˣ} : r⁻¹ ∈ spectrum R ↑a⁻¹ → r ∈ spectrum R ↑a

Alias of the forward direction of spectrum.inv₀_mem_inv_iff.

@[simp]

theorem spectrum.zero_eq {𝕜 : Type u} {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] [Nontrivial A] : spectrum 𝕜 0 = {0}

Without the assumption Nontrivial A, then 0 : A would be invertible.

@[simp]

theorem spectrum.scalar_eq {𝕜 : Type u} {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] [Nontrivial A] (k : 𝕜) : spectrum 𝕜 ((algebraMap 𝕜 A) k) = {k}

@[simp]

theorem spectrum.one_eq {𝕜 : Type u} {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] [Nontrivial A] : spectrum 𝕜 1 = {1}

theorem spectrum.smul_eq_smul {𝕜 : Type u} {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] [Nontrivial A] (k : 𝕜) (a : A) (ha : (spectrum 𝕜 a).Nonempty) : spectrum 𝕜 (k • a) = k • spectrum 𝕜 a

the assumption (σ a).Nonempty is necessary and cannot be removed without further conditions on the algebra A and scalar field 𝕜.

theorem spectrum.nonzero_mul_comm {𝕜 : Type u} {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] (a b : A) : spectrum 𝕜 (a * b) \ {0} = spectrum 𝕜 (b * a) \ {0}

theorem spectrum.map_inv {𝕜 : Type u} {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] (a : Aˣ) : (spectrum 𝕜 ↑a)⁻¹ = spectrum 𝕜 ↑a⁻¹

theorem AlgHom.mem_resolventSet_apply {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Ring A] [Algebra R A] [Ring B] [Algebra R B] [FunLike F A B] [AlgHomClass F R A B] (φ : F) {a : A} {r : R} (h : r ∈ resolventSet R a) : r ∈ resolventSet R (φ a)

theorem AlgHom.spectrum_apply_subset {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Ring A] [Algebra R A] [Ring B] [Algebra R B] [FunLike F A B] [AlgHomClass F R A B] (φ : F) (a : A) : spectrum R (φ a) ⊆ spectrum R a

theorem AlgHom.apply_mem_spectrum {F : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [Ring A] [Algebra R A] [FunLike F A R] [AlgHomClass F R A R] [Nontrivial R] (φ : F) (a : A) : φ a ∈ spectrum R a

@[simp]

theorem AlgEquiv.spectrum_eq {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] [EquivLike F A B] [AlgEquivClass F R A B] (f : F) (a : A) : spectrum R (f a) = spectrum R a

@[simp]

theorem spectrum.units_conjugate {R : Type u_1} {A : Type u_2} [CommSemiring R] [Ring A] [Algebra R A] {a : A} {u : Aˣ} : spectrum R (↑u * a * ↑u⁻¹) = spectrum R a

Conjugation by a unit preserves the spectrum, inverse on right.

@[simp]

theorem spectrum.units_conjugate' {R : Type u_1} {A : Type u_2} [CommSemiring R] [Ring A] [Algebra R A] {a : A} {u : Aˣ} : spectrum R (↑u⁻¹ * a * ↑u) = spectrum R a

Conjugation by a unit preserves the spectrum, inverse on left.