$p$-th matricial numerical range

Definition

The $p$-th matricial numerical range is the special case of $(p,k)$-numerical range for $k=1$.

Let $M_n$ will be the set of all matrices of dimension $n$ and by ${\mathrm{U}}_n$ it will be denoted the set of all unitary matrices of dimension $n$. The $p$-th numerical range (also known as matricial numerical range [1]) of the matrix $A\in M_n$ is defined as [2]

$$W(p:A):=\{X^{†}AX:C\in {\mathbb{C}}^{n\times p},X^{†}X={\mathbb{1}}_p\}.$$

We can observe that $W(n:A)=\{U^{A}U:X_n\}$ and $W(1:A)=W(A)$ as standard numerical range. In particular, many generalizations of standard numerical range $W(A)$ are actually the ranges of certain scalar-valued functions on $W(p:A)$ (see [3], [4]) so it is worthwhile to study the properties of $W(p:A)$.

Convexity of $W(p:A)$

Let $A\in M_n$ will be any matrix of dimension $n$, then in general case the set $W(p:A)$ is non-convex [5]. The following theorems (see [2]) give us the conditions to matrix $A$ and its eigenvalues so as to the set $W(p:A) will be convex.

Theorem

Let $A\in M_n$ will be Hermitian matrix with eigenvalues ${\lambda }_1\ge \dots \ge {\lambda }_n$. The set $W(p:A)$ is convex if and only if ${\lambda }_1={\lambda }_p$ and ${\lambda }_{n-p+1}={\lambda }_n$.

We can see that if $A\in M_n$ is Hermitian and $p>n/2$, when $W(p:A)$ is convex if and only if $A$ is a scalar matrix.

We can see other theorems involving convexity of $W(p:A)$ in [6].

Properties

Let $1<p<n$.

- If all $X\in W(p:A)$ are scalar matrices, then $A$ is a scalar matrix. Opposite implication is obvious;

- All $X\in W(p:A)$ are Hermitian if and only if $A$ is Hermitian;

- All $X\in W(p:A)$ are normal if and only if $A$ is essentially Hermitian.

References

1. [1]W.-F. Chuan, “The unitary equivalence of compact operators,” Glasgow Mathematical Journal, vol. 26, no. 2, pp. 145–149, 1985, [Online]. Available at: https://www.cambridge.org/core/journals/glasgow-mathematical-journal/article/unitary-equivalence-of-compact-operators/799971338C614A5088FB71FE0691659A.
   https://www.cambridge.org/core/journals/glasgow-mathematical-journal/article/unitary-equivalence-of-compact-operators/799971338C614A5088FB71FE0691659A
2. [2]C.-K. Li and N.-K. Tsing, “On the k th matrix numerical range,” Linear and Multilinear Algebra, vol. 28, no. 4, pp. 229–239, 1991, [Online]. Available at: https://www.tandfonline.com/doi/abs/10.1080/03081089108818047.
   https://www.tandfonline.com/doi/abs/10.1080/03081089108818047
3. [3]C.-K. Li, B.-S. Tam, and N.-K. Tsing, “Linear operators preserving the (p, q)-numerical range,” Linear Algebra and its Applications, vol. 110, pp. 75–89, 1988, [Online]. Available at: https://www.sciencedirect.com/science/article/pii/0024379583901337.
   https://www.sciencedirect.com/science/article/pii/0024379583901337
4. [4]C.-K. Li and N.-K. Tsing, “The numerical range of derivations,” Linear Algebra and its Applications, vol. 119, pp. 97–119, 1989, [Online]. Available at: https://core.ac.uk/download/pdf/82777226.pdf.
   https://core.ac.uk/download/pdf/82777226.pdf
5. [5]R. C. Thompson, Research problem the matrix numerical range. Taylor and Francis, 1987.
   https://www.tandfonline.com/doi/abs/10.1080/03081088708817807
6. [6]P.-S. Lau, C.-K. Li, Y.-T. Poon, and N.-S. Sze, “Convexity and star-shapedness of matricial range,” Journal of Functional Analysis, vol. 275, no. 9, pp. 2497–2515, 2018, [Online]. Available at: https://www.sciencedirect.com/science/article/pii/S0022123618301277.
   https://www.sciencedirect.com/science/article/pii/S0022123618301277