Hyperinvariant subspaces for some Bcircular operators
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We show that if A is a Hilbert-space operator, then the set of all projections onto hyperinvariant subspaces of A, which is contained in the von Neumann algebra v N (A) that is generated by A, is independent of the representation of vN(A), thought of as an abstract W*-algebra. We modify a technique of Foias, Ko, Jung and Pearcy to get a method for finding nontrivial hyperinvariant subspaces of certain operators in finite von Neumann algebras. We introduce the B-circular operators as a special case of Speicher's B-Gaussian operators in free probability theory, and we prove several results about a B-circular operator z, including formulas for the B-valued Cauchy- and R-transforms of z*z. We show that a large class of L([0, 1])-circular operators in finite von Neumann algebras have nontrivial hyperinvariant subspaces, and that another large class of them can be embedded in the free group factor L (F3). These results generalize some of what is known about the quasinilpotent DT-operator.
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