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numerical-range:generalizations:q-numerical-range

q-numerical range

Definition

Let $A$ be an $d \times d$ matrix. The q-numerical range of $A$ is the set: \[ W_q(A) = \left\{ z \in \mathbb{C}: z = \bra{x}A\ket{y}, |\braket{x}{y}| = q, \ket{x} \in \mathbb{C}^d, \ket{y} \in \mathbb{C}^d, \braket{x}{x} = 1, \braket{y}{y} = 1 \right\}. \]

Properties

Properties of $W_q(A)$ of a matrix $A$ of dimension $d \times d$ [1],[2]:

  1. Note that in the case $q=1$ we get back the standard numerical range;
  2. $W_q(A)$ is a convex and bounded set (Tsing theorem) [3];
  3. Unitarly invariant: $W_q(A) = W_q(UAU^\dagger)$ for any $U$ unitary matrix;
  4. Transpose invariant: $W_q(A) = W_q(A^\top)$;
  5. $W_{qz}(A) = zW_q(A)$ for any $z \in \mathbb{C}$, such that $|z|=1$;
  6. For any $\mu, \eta \in \mathbb{C}$, we have $W_q(\mu A + \eta \1) = \mu W_q(A) + \eta q$.
1. Mao-Ting Chien, Hiroshi Nakazato, 2006. The q-numerical range of a reducible matrix via a normal operator. Linear Algebra and its Applications, 419, North-Holland, pp.440–465.
2. Mao-Ting Chien, Hiroshi Nakazato, 2002. Davis–Wielandt shell and q-numerical range. Linear Algebra and its Applications, 340, Elsevier, pp.15–31.
3. Nam-Kiu Tsing, 1984. The constrained bilinear form and the C-numerical range. Linear Algebra and its Applications, 56, Elsevier, pp.195–206.
numerical-range/generalizations/q-numerical-range.txt · Last modified: 2019/03/01 16:28 by plewandowska