Spectrum reducing extension for one operator on a Banach space

Abstract

In this paper we show that, given an operator T T on a Banach space X X , there is an extension Y Y of X X such that T T extends in a natural way to an operator T ∼ {T^ \sim } on Y Y , and the spectrum of T ∼ {T^ \sim } is the approximate point spectrum of T 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 L containing elements g 1 {g_1} and g 2 {g_2} such that in no extension L ′ L’ of L L are the residual spectra of g 1 {g_1} and g 2 {g_{_2}} eliminated simultaneously.