Theorems · Theorem · functional analysis
spectrum.resolvent_zero_of_mem_spectrum
∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Ring A] [inst_2 : Algebra R A] {r : R} {a : A},
r ∈ spectrum R a → resolvent a r = 0- Defined in
- Mathlib.Algebra.Algebra.Spectrum.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 19 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringRingAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement · cited by 53,352
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Ringstatement and proof · cited by 7,463
- Algebra.algebraMapproof · cited by 4,706
- spectrumstatement and proof · cited by 510
- Ring.inverse_non_unitproof · cited by 26
- resolventstatement · cited by 23
- spectrum.mem_iffproof · cited by 13
Cited by2
Results whose statement or proof uses this declaration.
- MeasureTheory.measurable_resolventproof · cited by 2
- spectrum.mem_spectrum_iff_resolvent_zeroproof · cited by 0