The structure of an isometric tuple

An $n$-tuple of operators $(V_1,...,V_n)$ acting on a Hilbert space $H$ is said to be isometric if the operator $[V_1\...\ V_n]:H^n\to H$ is an isometry. We prove a decomposition for an isometric tuple of operators that generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.Comment: 30 pages; significant change

The Hopf structure of some dual operator algebras

We study the Hopf structure of a class of dual operator algebras corresponding to certain semigroups. This class of algebras arises in dilation theory, and includes the noncommutative analytic Toeplitz algebra and the multiplier algebra of the Drury-Arveson space, which correspond to the free semigroup and the free commutative semigroup respectively. The preduals of the algebras in this class naturally form Hopf (convolution) algebras. The original algebras and their preduals form (non-self-adjoint) dual Hopf algebras in the sense of Effros and Ruan. We study these algebras from this perspective, and obtain a number of results about their structure.Comment: 30 page

Spectral conditions on Lie and Jordan algebras of compact operators

We investigate the properties of bounded operators which satisfy a certain spectral additivity condition, and use our results to study Lie and Jordan algebras of compact operators. We prove that these algebras have nontrivial invariant subspaces when their elements have sublinear or submultiplicative spectrum, and when they satisfy simple trace conditions. In certain cases we show that these conditions imply that the algebra is (simultaneously) triangularizable.Comment: 14 page