2 ORIGINS IN SINGLE OPERATOR THEORY are Jt = {0} or Jt = ^f the term intransitive simply means not transitive. The following result is an unpublished theorem attributed to von Neumann. Á generalization to Banach Spaces was found by Aronszajn and Smith [4]. THEOREM 1.2. Every compact Operator on infinite-dimensional Hubert space is intransitive. Responding to a conjecture of Kennan Smith which was popularized by Paul Halmos, A. Bernstein and A. Robinson [17] found a significant generalization of Theorem 1.2 in which the hypothesis that Ô is compact is replaced by the hypothesis that p(T) is compact for some nontrivial polynomial p. Their proof had metamathematical aspects which made many functional analysts uncomfort- able, and soon Halmos published a somewhat improved " translation" of their proof into more conventional Operator theoretic terms [36]. The latter was generalized and simplified in [16], and in short order a flurry of papers had inundated the subject. Years later, Lomonosov found a dramatic generalization: if an Operator algebra jtf commutes with á nonzero compact Operator, then stf is intransitive. Lomonosov's method was entirely new, and seemingly, his result had rendered obsolete much of the preceding work on invariant subspaces. While that Statement is true in some limited sense, it is certainly misleading. And there is a lesson here. What has survived from the pre-Lomonosov methodol- ogy is a concept (quasitriangularity). This concept has suggested new problems, and new formulations of old problems, which have led to remarkable progress in single Operator theory and in Operator algebras. Let us begin by sketching the essential idea behind the proof in [16]. One is given a quasinilpotent Operator Ô such that p(T) Ö 0 for every polynomial/? Ö 0, and a cyclic vector î for T. Let Pn be the ^-dimensional projection onto [£, ã, Ô2æ,...,Ô~éî]. ç Pn is not invariant under Ã, but a routine computation shows that the sequence {Pn: ç 1} is asymptotically invariant in the following sense: (1.2) lim | | ( l - P j r P | | = 0, « - * oo [16, p. 61]. If the norm-closed algebra generated by 1 and Ô contains a nonzero compact Operator, then one can construct a nontrivial invariant subspace for T. Briefly, one finds a judiciously chosen sequence Qn of invariant projections for the sequence of finite-dimensional Operators One then extracts a subsequence Qn, Qn ,... which converges weakly to a positive selfadjoint Operator Q. If one makes the "right" choice of {Qn}, then it can be shown t h a t ^ = {£: ß | = î} is a nontrivial Ã-invariant subspace. Halmos made the condition (1.2) into a definition, thereby initiating the theory of quasitriangular Operators [37]. Let us recall the basic definitions. An Operator Ô G ££(jf) is called triangulär if there is an increasing sequence Pl ^ P2 · · ·
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