On some generalizations of the numerical range and their properties

For a bounded linear operator A (or, in the finite dimensional setting, an n-by-n matrix A) its classical numerical range is defined as the mage of the unit sphere under the mapping f(x)=(Ax,x). The normalized numerical range results when in this definition (Ax,x) is replaced by (Ax,x)/||Ax||. We will discuss some general properties of this set, and in particular provide its complete description in the cases when A is normal or n=2