On the hyperinvariant subspace problem. III
[Part I appeared in J. Funct. Anal. 219, No. 1, 134--142 (2005; Zbl 1066.47004); Part II appeared in Indiana Univ. Math. J. 54, No. 3, 743--754 (2005; Zbl 1071.47006).] Let \(H\) be a separable, infinite-dimensional, complex Hilbert space and \(L(H)\) the algebra of all bounded linear operators on \(H\). A closed linear subspace \(M \subset H\) is called a nontrivial hyperinvariant subspace for an operator \(T \in L(H)\) if \(\{0 \}\not=M \not= H\) and \(RM \subset M\) for each \(R\) in the commutant of \(T\). In Section 2 of this paper, the authors introduce an equivalence relation, less general than quasisimilarity, but more general than similarity, which nevertheless preserves hyperinvariant subspace lattices. In Section 3, they give an equivalence relation on \(L(H)\), weaker than similarity but stronger than hyperquasisimilarity, that not only preserves the hyperlattices of operators but preserves the lattices of invariant subspaces. In Section 4, they show that all operators discussed have a hyperlattice isomorphic to that of one of a special class of (BCP)-operators. In the final section, they deal with reduction to quasidiagonal (BCP)-operators.
scientific article; zbMATH DE number 2172989
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