ABSTRACT

Does every operator T on a separable Hilbert space H have a non-trivial
closed invariant subspace? This is the famous and still open “invariant
subspace problem” for operators on a Hilbert space. A natural
generalization of the problem is: Let M be a von Neumann algebra on
a Hilbert space H. Does every operator T in M have a non-trivial closed
invariant subspace K affiliated with M? (K is affiliated with M, if
the orthogonal projection on K belongs to M).

In the special case that M is a II

_{1}-factor (i.e. a infinite
dimensional von Neumann factor with a bounded trace), it turns out
that “almost all” operators in M have non-trivial closed
invariant subspaces affiliated with M. More precisely, this holds
for all operators in M for which L. G. Brown’s spectral distribution
measure for T is not concentrated at a single point of the complex
plane. The result is obtained in collaboration with Hanne Schultz
(preprint 2006), and it relies in a crucial way on Voiculescu’s
free probability theory. This result was previously known only for
special operators such as Voiculescu’s circular operator and
more generally for the DT-operators introduced in a paper in the American
Journal of Mathematics (2004) in collaboration with Ken Dykema.

There will be three talks on this subject. The first will be a
survey talk on this topic. In the second talk, I will go more into
the technical details, and in particular explain how free probability
enters in the proof of the main result. In the third talk, I will
focus on examples and applications.

Haagerup's research is in operator theory, and covers many subareas
in the subject which are currently very active - random matrices,
free probability, C*-algebras & applications to mathematical
physics. He is among the most significant contributors to the theory
at present and his research has critical impact in virtually all
areas of operator theory.

Haagerup is a professor at Odense University in Denmark. He has
been an organizer of many major conferences, including several at
Oberwolfach, MSRI, the Institut Mittag-Leffler and elsewhere. He
is a member of the Royal Danish Academy of Science and Letters and
of the Norwegian Academy of Science, and has been the Editor-in-Chief
of Acta Mathematica.

Speakers in the Distinguished Lecture Series (DLS) have made outstanding
contributions to their field of mathematics. The DLS consists of
a series of three one-hour lectures.

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