Block Numerical Range – A Localisation of the Spectrum

Abstract: 

The numerical range was first studied in 1918 as a useful tool for localising the spectrum of a linear operator on a Hilbert space. The convexity of this localisation limits how much information we can obtain about the spectrum. In 2002, the block numerical range was introduced as a better, not necessarily convex localisation of the spectrum.

In this talk, we will introduce the block numerical range of a bounded n × n block operator matrix on a separable Hilbert space and explore its ability to localise the spectrum under varying decompositions of the Hilbert space.