= Resolvent-generated commutative algebra
{title2=$\sigma_{A_T}(T)=\sigma_{\mathcal B(X)}(T)$}
The closed unital algebra generated by a bounded operator and all its resolvents is commutative, because the generators commute. It has exactly the original operator spectrum at that operator: all required inverses outside that spectrum were included, and an inverse in the smaller algebra is also an operator inverse. This permits commutative-algebra <linear functional> calculus without replacing the spectrum by the possibly larger spectrum of a polynomial-generated algebra.
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