Theorems · Theorem · operator theory

ContinuousLinearMap.eq_zero_of_forall_hasEigenvalue_eq_zero

∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  [CompleteSpace E] {T : E →L[𝕜] E},
  IsCompactOperator ⇑T → (↑T).IsSymmetric → ((∀ (μ : 𝕜), Module.End.HasEigenvalue (↑T) μ → μ = 0) ↔ T = 0)

Defined in

Mathlib.Analysis.InnerProductSpace.Spectrum

Cited by

1 results in Mathlib

Foundations

Depth 197 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpace