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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.
18/10/2012, 16:30 — 17:30 — Room P3.10, Mathematics Building
Juan Rodríguez, Universidade do Algarve
Factorization of rational matrix functions and difference equations
In the beginning of the XX century, Plemelj introduced the notion
of factorization of matrix functions. If is a
regular matrix-function on the unit circle , by (left)
factorization of relative to , we mean the
following representation: where with
, and
() is analytic and regular in
(). The matrix-function
factorization find applications in many fields like diffraction
theory, the theory of differential equations and the theory of
singular integral operators.However, only for a few classes of
matrices is known the explicit formulas for the factors of the
factorization. In our talk we will show a new method to obtain a
factorization of rational matrix-functions. The constructed method
is based on the relation between the general solution of an
homogeneous Riemann-Hilbert problem and a solution of a linear
system of difference equations with constant coefficients. We also
will provide some examples of factorization of rational matrix
function, constructed with the developed factorization procedure.