4 OR1GINS IN SINGLE OPERATOR THEORY (1.5) implies that the Operator 00 ê= Ó (i - ß*M(ß* - ö*-x) k = l is an absolutely convergent series of finite rank Operators, and is therefore compact. Moreover, the Operator Ô = Á - Ê is triangulär because Ê has been constructed so that {\-Qk)KQk = {l-Qk)AQk9 and hence (1 - Qk)TQk = 0, k = 1,2,— Therefore Á = Ô + Ê belongs to The above argument shows that if {Pn} is a sequence of finite rank projections which increases to 1 and Á is an Operator such that ||(1 Pn)APn\\ - 0, then Á has a decomposition (1.6) A = T+K as a compact perturbation of an Operator Ô which leaves an infinite subsequence [Pn } of {Pn} invariant. Á natural question here is whether or not Ô and Ê can be chosen so that Ô leaves the entire sequence {Pn} invariant. The answer turns out to be yes but the proof is quite different from the proof of Theorem 1.3, and involves Operator algebraic techniques which will be discussed in Lecture 2 (Theorem 2.10). We will return to quasitriangular Operators presently. I want to digress for a time in order to describe some ways of thinking about compact perturbations that have been useful to me. Consider first the space l°° of all bounded complex-val- ued sequences á = {an: ç = 1,2,...}. Æ0 0 is a commutative C*-algebra with unit relative to the usual norm and Operations: for instance, the product of two sequences á and b is the sequence {anbn: ç = 1,2,...}. What does it mean for two sequences to have the "same properties"? We mean by this that there should be an automorphism of the given C*-algebra structure of l°° which carries one sequence to the other. It is not hard to determine the group of all automorphisms of /°°. The most general automorphism á is given by a permutation ð of the set Í of positive integers as follows: (1.7) á(á)={áð{ç):ç£Í} 9 á e ð. The reason is that an automorphism á of /°° must permute the minimal projec- tions of /°°, the latter are identified as the characteristic functions of singleton subsets of N, and thus we obtain a permutation ð of Í satisfying (1.7). What does it mean for two sequences á and b to have the "same asymptotic properties"? It is reasonable to require this to mean that there should be an automorphism á of l°° such that a{a) - b vanishes at infinity. Thus one may introduce an equivalence relation as follows: DEFINITION 1.8. á - b iff there is á permutation m o/N such that lim k(n)-^| =0 · 77-0 0
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