# $C$-numerical range

## Definition

Let $A$ be an $d \times d$ matrix and $(c_1, c_2, \ldots, c_d)$ be a real d-tuple. The $c$-numerical range of $A$ is the set \begin{equation} W_c(A) = \{ \sum_{j=1}^d c_j \bra{x_j} A \ket{x_j}: \{\ket{x_i}\}_{i=1}^d \mathrm{\;forms\; an\; orthonormal\; basis\; of\;} \mathbb{C}^d \}. \end{equation}

Let $C=\mathrm{\;diag}(c_1, c_2, \ldots, c_d)$. Then $\mu \in W_c(A) \iff \mu = \tr(CU^\dagger AU)$ for some unitary matrix $U$. This fact motivates to define $C$-numerical range by \begin{equation} W_C(A) = \{ \tr(CU^\dagger AU): \mathrm{ \;U \; unitary\; }\} \end{equation} for any square matrix $C$. For convenience one may define the unitary similarity orbit of matrix $A$ given by the formula \begin{equation} \mathcal{U}(A) = \{U^\dagger AU :\mathrm{\;U \; unitary\; }\}. \end{equation}

## Properties

Properties of $W_C(A)$ of a matrix $A$ of dimension $d \times d$:

1. $W_C(A)$ is a compact set;
2. Symmetry: $W_C(A) = W_A(C)$;
3. For any matrices $A, B$ of dimension $d \times d$, $W_C(A+B) \subseteq W_C(A) + W_C(B)$;
4. For any $\mu, \eta \in \mathbb{C}$, then $W_C(\mu A + \eta \1) = \mu W_C(A) + \eta \tr(C)$;
5. $W_C(U^\dagger AU) = W_C(A)$ for any unitary matrix $U$;
6. $W_{V^\dagger CV}(A) = W_C(A)$ for any unitary matrix $V$;
7. $W_C(A) = W_C(A^\top) \iff C^\top \in \mathcal{U}(C)$
8. $\overline{W_C(A) }= W_C(\overline{A}) \iff \overline{C} \in \mathcal{U}(C)$
9. $\overline{W_C(A) }= W_C(A^\dagger) \iff C^\dagger \in \mathcal{U}(C)$
10. If $c=[1,0,\ldots,0]$, then $W_c(A)$ reduces to the classical numerical range of $A$ and if $C=\mathrm{\;diag}(1,0,\ldots,0)$, then $W_C(A)$ reduces to the classical numerical range of $A$.

## Convexity

Convexity of $W_c(A)$ and $W_C(A)$ of a matrix $A$ of dimension $d \times d$:

• $W_c(A)$ is a convex set (Westwick theorem): 
• $W_C(A)$ is a convex set if one of the following holds: 
• there exist $\mu$, $\eta \in \mathbb{C}$ with $\mu \neq 0$ such that $\mu C+ \eta \1$ is hermitian;
• there exist $\mu \in \mathbb{C}$ such that $C - \mu \1$ is unitarity similar to $(C_{ij})_{1 \le i,j \le d}$ in block form, where $C_{ii}$ are square matrices and $C_{ij} = 0$ if $i \neq j+1$;
• there exist $\mu \in \mathbb{C}$ such that $C - \mu \1$ has rank one.

## Generalization

We can generalize the $C$-numerical range $W_C(A)$ to Schatten-class operators i.e. to $C\in\mathcal B^p(\mathcal H)$ and $A\in\mathcal B^q(\mathcal H)$ with condition $1/p + 1/q = 1$, and show that its closure is always star-shaped with respect to the origin .

Let $\mathcal{X}, \mathcal{Y}$ denote an arbitrary infinite-dimensional separable complex Hilbert space. Moreover, $\mathcal B(\mathcal X,\mathcal Y)$, $K( \mathcal{X},\mathcal{Y})$ and $B^p(\mathcal{X},\mathcal{Y})$ denote the set of all bounded, compact and $p$-th Schatten-class operators between $\mathcal X$ and $\mathcal Y$, respectively. By $\mathcal B^p( \mathcal{X}, \mathcal{Y})$ we denote all $p$-Schatten-class operators defined by \begin{equation} \mathcal{B}^p(\mathcal{X}, \mathcal{Y}) \coloneqq \{ C \in \mathcal{K}(\mathcal{X}, \mathcal{Y}) | \sum_{n=1}^\infty s_n(C) ^p < \infty \} \end{equation}

for $p [1,\infty )$ whereas the Schatten-$p$-norm of matrix $A$ $$\| A \|_p \coloneqq \left( \sum_{n=1}^\infty s_n(A)^p \right)^{1/p},$$ where sequence $(s_n)_{n=1}^{\infty}$ comes from above well-know Schmidt decomposition theorem.

### Schmidt decomposition

For each $C \in \mathcal K(\mathcal X,\mathcal Y)$, there exists a decreasing null sequence $(s_n(C)){n\in\mathbb N}$ in $[0,\infty)$ as well as orthonormal systems $(f_n){n \in \mathbb{N}}$ in $\mathcal{X}$ and $(g_n)_{n\in \mathbb{N}}$ in $\mathcal{Y}$ such that where the series converges in the operator norm.

Moreover, the sequence $(s_n(C))_{n\in\mathbb N}$ is uniquely determined by $C$.

### Definition

Let $p,q\in [1,\infty]$ be conjugate. Then for $C\in\mathcal B^p(\mathcal H)$ and $A\in\mathcal B^q(\mathcal H)$, we define the $C$-numerical range of $T$ to be

$W_C (A)\coloneqq \lbrace \operatorname{tr}(CU^\dagger AU)\,|\,U\in\mathcal B(\mathcal H)\text{ unitary}\rbrace\,.$

The properties of $C$-numerical range in infinite-dimensional vector space for Schatten-class operator we can find in .

1. R. Westwick, “A theorem on numerical range,” Linear and Multilinear Algebra, vol. 2, no. 4, pp. 311–315, 1975, [Online]. Available at: https://www.tandfonline.com/doi/abs/10.1080/03081087508817074?journalCode=glma20.
2. C.-K. Li, “C-numerical ranges and C-numerical radii,” Linear and Multilinear Algebra, vol. 37, no. 1-3, pp. 51–82, 1994, [Online]. Available at: https://www.tandfonline.com/doi/abs/10.1080/03081089408818312?journalCode=glma20.
3. Y. Zhang and X. Fang, “c-numerical range of operator products on B (H),” arXiv preprint arXiv:1901.05245, vol. 0, 2019, [Online]. Available at: https://arxiv.org/abs/1901.05245.
4. G. Dirr and F. vom Ende, “The C-Numerical Range for Schatten-Class Operators,” arXiv preprint arXiv:1808.06898, vol. 0, 2018, [Online]. Available at: https://arxiv.org/abs/1808.06898.