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numerical-range:generalizations:p-th-matricial-range

$p$-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]. \begin{equation} W(p:A) := \left\{ X^\dagger A X: X \in \mathbb{C}^{n \times p}, X^\dagger X = \mathbf{1}_p\right\} \end{equation}

We can observe that $W(n:A) = \left\{ U^\dagger A U: X \in \mathrm{U}_n \right\}$ 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 $\lambda_1 \ge \ldots \ge \lambda_n$. 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.

The example of application of $p$-th matricial numerical range we can see (for example) [7].

1. Wai-Fong Chuan, 1985. The unitary equivalence of compact operators. Glasgow Mathematical Journal, 26, Cambridge University Press, pp.145–149.
2. Chi-Kwong Li, Nam-Kiu Tsing, 1991. On the k th matrix numerical range. Linear and Multilinear Algebra, 28, Taylor and Francis, pp.229–239.
3. Chi-Kwong Li, Bit-Shun Tam, Nam-Kiu Tsing, 1988. Linear operators preserving the (p, q)-numerical range. Linear Algebra and its Applications, 110, Elsevier, pp.75–89.
4. Chi-Kwong Li, Nam-Kiu Tsing, 1989. The numerical range of derivations. Linear Algebra and its Applications, 119, North-Holland, pp.97–119.
5. Robert C Thompson, 1987. Research problem the matrix numerical range. Taylor and Francis.
6. Pan-Shun Lau, Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze, 2018. Convexity and star-shapedness of matricial range. Journal of Functional Analysis, 275, Elsevier, pp.2497–2515.
numerical-range/generalizations/p-th-matricial-range.txt · Last modified: 2020/05/13 22:20 by rkukulski