ORIGINS IN SINGLE OPERATOR THEORY 7 spectrum (namely the unit circle), whereas both A and B are Fredholm operators whose Fredholm indices are, respectively, - 1 , and +1. It follows that A and B cannot be equivalent. Let us return now to quasitriangular operators. In 1968, Halmos showed [37] that there exist operators which are not quasitriangular (the shift is one such). Douglas and Pearcy later clarified the situation somewhat by proving [28]. THEOREM 1.16. If A is quasitriangular, then index(/4 - XI) 0 for every A G C for which A - XI is semi-Fredholm, This implies Halmos' earlier result, because the shift is a Fredholm operator whose index is - 1 . In a series of papers which contain a deep analysis of the spectral properties of operators, Apostol, Foias, and Voiculescu provided a remarkable converse to Theorem 1.16 [3, Corollary 5.5]. THEOREM 1.17. If A is an operator such that index(^ - XI) ^ 0 for every A G C for which A — XI is semi-Fredholm, then A is quasitriangular. Thus, the only obstruction to membership in the class 2,3~ is an index obstruction. Theorem 1.17 also has implications about invariant subspaces it implies that every nonquasitriangular operator is intransitive. For by Theorem 1.17, such an operator A would have a scalar translate A - XI of negative index, hence A* - XI would have an eigenvector, hence A would have an invariant subspace of codimension one. So if there exists a transitive operator on a Hilbert space (and I personally believe that such operators do exist), then it must be quasitriangular.
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