Talk:Resolvent Set of Element of Banach Algebra is Open

I realised this. There's no problem with saying that $\map {\rho_A} x = \map {\rho_{A_+} } {\tuple {x, 0} }$ if $A$ is not unital right? This is just the complement of the set we've defined to be the spectrum in non-unital algebras. Not in the text I have so would search before. Caliburn (talk) 20:58, 19 August 2023 (UTC)

I don't know if you should define the resolvent of $x$ in a non-unital algebra like this, since the resolvent $\paren {x - \lambda 1}^{-1}$ does not make sense anymore. I guess, one only talk about spectrum, but about resolvent. I didn't check it. --Usagiop (talk) 21:27, 19 August 2023 (UTC)

Sorry, I think you are simply right. No reason not to define it. --Usagiop (talk) 21:33, 19 August 2023 (UTC)