Edward Azoff: Invariant subspaces

This talk is one of the department's colloquium talks. They are pitched at a general mathematical audience, meaning that it's meant for the mathematically proficient (which often means that one should at least be a grad student in math), but not an expert in the particular area of the talk. Abstract: Let T be a bounded linear operator acting on a complex Hilbert space H. A closed subspace M of H is said to be invariant under T if T(M) is contained in M. The trivial subspaces {0} and H itself are invariant under every operator. The invariant subspace problem, open for over 60 years, asks whether there are always others. This talk will briefly survey progress on the question, ranging from positive results for special classes of operators to theorems and counterexamples relating to generalized versions of the problem. One focus will be on the deep connections with analytic function theory first developed by A. Beurling. Tools coming from other areas of mathematics, including fixed point theory, auxiliary topologies, and duality will also be featured. The UGA Math Club extends its warm thanks to Jon Hanke for helping us with both the hardware and software of making our videos.