Elements of spectral radius zero

The spectral radius of an element vanishes exactly when its quasispectrum, or in a unital algebra its spectrum, is contained in {0}. In a complex Banach algebra these are the quasinilpotent elements, by Gelfand's formula. Both statements are algebraic: they need a norm only on the scalars, and no completeness.

For the same reason a nonunital algebra homomorphism, with no continuity assumption, does not increase spectral radii: it can only shrink quasispectra.

theorem TauCeti.spectralRadius_eq_zero_iff {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NonUnitalRing A] [Module 𝕜 A] {a : A} : spectralRadius 𝕜 a = 0 ↔ quasispectrum 𝕜 a ⊆ {0}

The spectral radius of an element vanishes exactly when its quasispectrum is contained in {0}.

theorem TauCeti.spectralRadius_eq_zero_iff_spectrum_subset_singleton_zero {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [Ring A] [Algebra 𝕜 A] {a : A} : spectralRadius 𝕜 a = 0 ↔ spectrum 𝕜 a ⊆ {0}

In a unital algebra, the spectral radius of an element vanishes exactly when its spectrum is contained in {0}.

theorem NonUnitalAlgHom.spectralRadius_apply_le {F : Type u_1} {𝕜 : Type u_2} {A : Type u_3} {B : Type u_4} [NormedField 𝕜] [NonUnitalRing A] [NonUnitalRing B] [Module 𝕜 A] [Module 𝕜 B] [FunLike F A B] [NonUnitalAlgHomClass F 𝕜 A B] (φ : F) (a : A) : spectralRadius 𝕜 (φ a) ≤ spectralRadius 𝕜 a

A nonunital algebra homomorphism, not assumed continuous, does not increase spectral radii, since it can only shrink quasispectra.