Talk by Marco Marletta

The numerical range of an operator $T$ in a Hilbert space is a set in the complex plane, given by all complex numbers of the form $(Tu,u)$, where $u$ ranges over all unit vectors in the domain of $T$. This set, usually denoted $W(T)$, is convex, and if the
spectrum of $T$ is not the whole complex plane then it will be contained in the closure of $W(T)$. These basic facts lie behind many methods for estimating whether or not some PDE is solvable, by getting estimates on where the spectrum of an associated operator lies.

The essential numerical range is a generally smaller set, still convex, which excludes all eigenvalues of $T$ of finite multiplicity, and is intended to capture the essential spectrum of $T$. It was studied extensively in the 1960s for Calkin algebras of operators (and hence, in particular, only for bounded operators).

In this talk, based on joint work with Sabine Boegli and Christiane Tretter, I shall speak about what happens when one considers unbounded operators. Many unexpected pathologies emerge and can be illustrated even with very simple diagonal operators in $\ell^2$ spaces.

If time permits I shall also discuss some results for operator pencils $T - \lambda B$, where the numerical range is no longer convex. This lack of convexity is immensely helpful in many applications, including the Dirac equation.