This is an introduction to the notion of numerical range for bounded linear operators on Hilbert space. The main results are: determination of the numerical range for two by two matrices, the Toeplitz-Hausdorff Theorem establishing the convexity of the numerical range for any Hilbert space operator, and a detailed discussion of the relationship between the numerical range and the spectrum. The ...

The classical numerical range of a quadratic operator is an elliptical disk. This result is extended to different kinds of generalized numerical ranges. In particular, it is shown that for a given quadratic operator, the rank-k numerical range, the essential numerical range, and the q-numerical range are elliptical disks; the c-numerical range is a sum of elliptical disks, and the Davis-Wieland...

We prove an ellitpical range theorem for the numerical range associated with SO(3). The geometry behind the elliptical range theorem is explained. The result is analogous to the well-known elliptical range theorem of the C-numerical range associated with U(2) or SU(2). We then apply the result to give some necessary conditions of the sharp points of the C-numerical range assoicated with SO(n) w...