The numerical range: a survey of classical results and some recent developments
Ilya M. Spitkovsky, College of William and Mary (USA) and New York University Abu Dhabi (United Arab Emirates) (cartel)
Viernes 13 de junio, ICMAT, Aula Naranja, 11:30
\[\]The numerical range (also know as the field of values, or the Hausdorff set) of a matrix
\[ A\]or, more generally, of a bounded linear operator acting on a Hilbert space, is defined as the range of the associated quadratic form
\[(Ax,x)\]on the unit sphere. It was introduced almost 100 years ago in pioneering works by Toeplitz and Hausdorff and has been an active area of research ever since. We will discuss briefly its basic classical properties (elliptical range theorem, convexity, spectrum containment), and will after that concentrate on the following:
- Possible shapes of the numerical range in finite dimensional case (which is actually a harder question than its infinite dimensional version), including persistence of ellipticity in higher dimensions and flat portions phenomenon on the boundary;
- Continuity properties of the inverse generating function, that is, the (multivalued) inverse of the mapping \[ x\to (Ax,x)\]. The notions of strong and weak continuity will be introduced, and criteria for each of them to hold/fail established, both in terms of the geometry of the numerical range and analytic properties of the associated eigenfunctions.