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numerical-range:generalizations:c-numerical-range

$C$-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 $$ W_c(A) = \left\{ \sum_{j=1}^d c_j \bra{x_j} A \ket{x_j}: \{\ket{x_i}\}_{i=1}^d \mathrm{\;forms\; an\; orthonormal\; basis\; of\;} \mathbb{C}^d \right\}. $$ 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 $$ W_C(A) = \left\{ \tr(CU^\dagger AU): \mathrm{ \;U \; unitary\; }\right\} $$ for any square matrix $C$. For convenience one may define the unitary similarity orbit of matrix $A$ given by the formula $$ \mathcal{U}(A) = \left\{U^\dagger AU :\mathrm{\;U \; unitary\; }\right\}. $$

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 $d \times d$:

  1. $W_c(A)$ is a convex set (Westwick theorem): [1]
  2. $W_C(A)$ is a convex set if one of the following holds: [2]
    • 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 [3].

Let $\mathcal{X}, \mathcal{Y}$ denote an arbitrary infinite-dimensional separable complex Hilbert space. Moreover, $\mathcal B(\mathcal X,\mathcal Y)$, $\mathcal K(\mathcal X,\mathcal Y)$ and $\mathcal B^p(\mathcal X,\mathcal Y)$ 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) := \Big\lbrace C \in\mathcal K(\mathcal X,\mathcal Y)\,\Big|\,\sum\nolimits_{n=1}^\infty s_n(C)^p<\infty\Big\rbrace \end{equation} for $p\in [1,\infty)$ whereas the Schatten-$p$-norm of matrix $A$ \begin{equation} ||A||_p := \Big(\sum_{n=1}^\infty s_n(A)^p\Big)^{1/p} \end{equation} 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 \begin{align*} C = \sum_{n=1}^\infty s_n(C)\langle f_n,\cdot\rangle g_n\,, \end{align*} 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 \emph{$C$-numerical range} of $T$ to be \begin{equation} W_C (A):=\lbrace \operatorname{tr}(CU^\dagger AU)\,|\,U\in\mathcal B(\mathcal H)\text{ unitary}\rbrace\,. \end{equation}

The properties of $C$-numerical range in infinite-dimensional vector space for Schatten-class operator we can find in [4].

1. R Westwick, 1975. A theorem on numerical range. Linear and Multilinear Algebra, 2, Taylor & Francis, pp.311–315.
2. Chi-Kwong Li, 1994. C-numerical ranges and C-numerical radii. Linear and Multilinear Algebra, 37, Taylor & Francis, pp.51–82.
3. Yanfang Zhang, Xiaochun Fang, 2019. c-numerical range of operator products on B (H). arXiv preprint arXiv:1901.05245.
4. Gunther Dirr, Frederik vom Ende, 2018. The C-Numerical Range for Schatten-Class Operators. arXiv preprint arXiv:1808.06898.
numerical-range/generalizations/c-numerical-range.txt · Last modified: 2020/05/16 10:42 by plewandowska