Taibleson Colloquium: Finding ellipses: Blaschke products and the numerical range

Abstract: The numerical range of an n × n complex matrix A is defined by
W(A) = {< Ax, x >: x ∈ Cⁿ, ||x|| = 1}.
In general, it’s not easy to compute the shape of the numerical range. In
this talk, we investigate the question of when numerical ranges of matrices
are elliptical by connecting this phenomenon to two seemingly different
settings: function theory and projective geometry. Starting with n = 2
and extending to general n leads to a class of operators known as compressions of the shift operator. This viewpoint provides new insight into the
numerical ranges of these operators and highlights special features that
emerge when the numerical range is an ellipse.

Host: John McCarthy

Reception to follow at Cupples I, Room 200 (Lounge) from 2:00pm to 3:00pm