- #1

AxiomOfChoice

- 533

- 1

[itex]\lambda \in \sigma_r(A)[/itex] iff [itex](\forall \psi \in H, \psi \neq 0)((\lambda - A) \psi \neq 0)[/itex] iff [itex]\ker (\lambda - A) = \{0\}[/itex] iff [itex]\lambda - A[/itex] is injective?

So isn't the condition that [itex]\lambda[/itex] isn't an eigenvalue equivalent to the condition [itex]\lambda - A[/itex] is injective?