Spectrum reducing extension for one operator on a Banach space

Abstract:

In this paper we show that, given an operator $T$ on a Banach space $X$, there is an extension $Y$ of $X$ such that $T$ extends in a natural way to an operator ${T^ \sim }$ on $Y$, and the spectrum of ${T^ \sim }$ is the approximate point spectrum of $T$. This answers a question posed by Bollobás, and contributes to a theory investigated by Shilov, Arens, Bollobás, etc. The unusual transfinite construction is similar to that which we used earlier to find an inverse producing extension for a commutative unital Banach algebra which eliminates the residual spectrum of one element. We also give a counterexample, consisting of a Banach algebra $L$ containing elements ${g_1}$ and ${g_2}$ such that in no extension $L’$ of $L$ are the residual spectra of ${g_1}$ and ${g_{_2}}$ eliminated simultaneously.