Spectrum Reducing Extension for one Operator on a Banach Space TeX

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 $T^∼$ on $Y$, and the spectrum of $T^∼$ is the approximate point spectrum of $T$. This answers a question posed by Bollobas, and contributes to a theory investigated by Shilov, Arens, Bollobas, 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.