22/11/2012, 16:30 — 17:30 — Room P3.10, Mathematics Building
Janko Bracic, University of Ljubljana, Slovenia
Numerical ranges and hyperreflexivity
Let be a complex Hilbert space and let
be the
unit sphere of . Every bounded linear operator on
defines a quadratic form as follows where denotes the inner product of .
The image of is the numerical range
of . It is not hard to see that the spectrum of an operator is a
subset of the closure of the numerical range, which means that
numerical ranges are a useful tool in locating the spectrum. Some
classical results about numerical ranges will be presented; for
instance, the Toeplitz-Hausdorff Theorem and the Hildebrandt's
Theorem. The hyperreflexivity of sets of operators determined by
the numerical range will be discussed, as well.