Invariant subspaces of Voiculescu's circular operator

On density of transitive operator algebras

“…We can get a similar result as in [3] if such a condition on the spectra can be replaced by a norm condition. Proof.…”

“…If T has the upper triangular form satisfying the condition on the spectra as in Proposition 3.1 in [3], then the subspace H r is a non-trivial invariant subspace for T . We can get a similar result as in [3] if such a condition on the spectra can be replaced by a norm condition.…”

“…From Voiculescu's free probability theory, we see that the joint distribution of a circular system can be approximated by the joint distributions of corresponding systems of independent Gaussian random matrices. Dykema and Haagerup [3] found that Voiculescu's matrix model leads to upper triangular models for the circular operator and the DT-operator. Using upper triangular realizations of the circular free Poisson element, they prove that the circular operator and each circular free Poisson operator which arise naturally in the free probability theory have a continuous family of invariant subspaces relative to the von Neumann algebra which it generates.…”

“…This paper is concerned with the transitive algebra question and the invariant subspace problem relative to von Neumann algebras [3,6]. Many theorems and open problems concerning the invariant subspace problem and the transitive algebra question remain to be solved.…”

“…In Section 2, we prove that the algebra generated by the set {λ a 1 + λ a 2 , λ a 3 + λ a 4 } and the commutant R(F ∞ ) of L(F ∞ ) is strong-operator dense in B(H) where a 1 , a 2 , a 3 and a 4 are four of generators of F ∞ . In the third section, we explicitly construct the invariant subspace of some operator using a similar model as upper triangular matrix models for circular free Poisson operators [3]. Furthermore, the relation between the transitive algebra question and the invariant subspace problem relative to finite von Neumann algebras is discussed with some remarks and examples.…”