If [itex]A[/itex] is a bounded operator on a Hilbert space [itex]H[/itex], isn't the following true of the residual spectrum [itex]\sigma_r(A)[/itex]:(adsbygoogle = window.adsbygoogle || []).push({});

[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?

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# Definition of the residual spectrum

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