Theorems · Theorem · functional analysis

spectrum.resolvent_sub_resolvent

∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Ring A] [inst_2 : Algebra R A] {a b : A} {r : R},
  r ∈ resolventSet R a → r ∈ resolventSet R b → resolvent a r - resolvent b r = resolvent a r * (a - b) * resolvent b r

The second resolvent identity: for r in the resolvent set of both a and b, resolvent a r - resolvent b r = resolvent a r * (a - b) * resolvent b r.

Defined in

Mathlib.Algebra.Algebra.Spectrum.Basic

Cited by

0 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringRingAlgebra