Jim Haley, Pure Mathematics, University of Waterloo
"Strongly Reductive Operators and Operator Algebras"
An operator is reductive if every invariant subspace is also invariant for the adjoint of the operator. The invariant subspace problem is equivalent to the question of whether all reductive operators are normal. Strongly reductive operators are reductive operators that satisfy an asymptotic form of reductivity: every "almost" invariant subspace for an operator is "almost" invariant for the operator's adjoint. We will present a number of results for strongly reductive operators, including the fact that strongly reductive operators are normal.
We can also define an analogue of strong reductivity for operator algebras in terms of subspaces that are invariant for all operators in the algebra. We will discuss a result of Prunaru that states that closed strongly reductive operator algebras are self-adjoint.