Normalized numerical range

Definition

The classical numerical range $F(A)$ of $A\in M_n(\mathbb{C})$ is by definition the set of values of the corresponding quadratic form $x^{\ast }Ax$ on the unit sphere $S^n:=\{x^n:\Vert x\Vert =1\}$of${}^n$. Equivalently, \[ F(A) =\{(x^*Ax)/||x||^{2 x}n\{0\}\}. \] Various modifications and generalization of the numerical range have been considered in the literature. Our paper is concerned with the so called normalized numerical range. Defined as $$F_N(A):=\{x^n,Ax0\},$$ it was introduced in [1], and then further investigated in for example [2].

Properties

Suppose that $A\in M_n(\mathbb{C})$. Then:

• For all $z\in F_N(A)$, $|z|\le 1$.

*If $z\in F_N(A)$, then $|z|=1$ if and only if $z=/||$for some$(A)$.

• $F_N(A)$ is unitarily invariant: $F_N(U^{\ast }AU)=F_N(A)$ for any unitary $U\in M_n(\mathbb{C})$.

• $F_N(e^{i\theta }A)=e^{i\theta }{F}_N(A)$ for all $[0,2)$.

• $F_N(cA)=F_N(A)$ for all $c>0$.

• If $A$ is invertible, then $F_N(A)$ is closed.

The following theorems and examples are taken from [3].

Theorem 1

Suppose that $A\in M_2(\mathbb{C})$ has non-zero eigenvalues ${\lambda }_1,{\lambda }_2$ such that ${\lambda }_1/{\lambda }_2<0$. Then $F_N(A)$ is a closed elliptical disk. In the case when $A>0$, the ellipse is given by the equation

Theorem 2

For $A\in M_2(\mathbb{C})\mathrm{\setminus }\{0\}$ with eigenvalues ${\lambda }_1$ and ${\lambda }_2$, the boundary of $F_N(A)$ is an ellipse if and only if $|{\lambda }_1|=|{\lambda }_2|$ or ${\lambda }_1/{\lambda }_2<0$.

Examples

1. The normalized numerical range of $A =

$.

2. The normalized numerical range of $B =

$.

3. The normalized numerical range of $C =

$.

For matrix $A$ is a typical example of convex normalized numerical range that is not an ellipse. The normalized numerical range of matrix $B$ is not convex, but has a smooth boundary (boundary is differentiable). Finally, the last example for matrix $C$ has a boundary that is not smooth at one point. All three of these examples have boundaries that satisfy irreducible 8th degree polynomial equations.

References

1. [1]W. Auzinger, “Sectorial operators and normalized numerical range,” Applied numerical mathematics, vol. 45, no. 4, pp. 367–388, 2003, [Online]. Available at: https://www.sciencedirect.com/science/article/pii/S0168927402002544.
   https://www.sciencedirect.com/science/article/pii/S0168927402002544
2. [2]L. Z. Gevorgyan, “Normalized numerical ranges of some operators,” Operators and Matrices, vol. 3, no. 1, pp. 145–153, 2009, [Online]. Available at: https://nyuscholars.nyu.edu/en/publications/on-the-normalized-numerical-range.
   https://nyuscholars.nyu.edu/en/publications/on-the-normalized-numerical-range
3. [3]B. Lins, I. M. Spitkovsky, and S. Zhong, “The normalized numerical range and the Davis–Wielandt shell,” Linear Algebra and its Applications, vol. 546, pp. 187–209, 2018, [Online]. Available at: https://www.sciencedirect.com/science/article/pii/S0024379518300417.
   https://www.sciencedirect.com/science/article/pii/S0024379518300417