# Numerical Shadow

The web resource on numerical range and numerical shadow

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numerical-range:generalizations:maximal-numerical-range

# Maximal numerical range

## Definition

Let $A$ be an $d \times d$ matrix. The maximal numerical range of $A$ is the set: $W_0(A) = \left\{ z \in \mathbb{C}: \bra{x_n}A\ket{x_n} \to z, \|A\ket{x_n} \| \to \|A\|, \braket{x_n}{x_n} = 1, \ket{x_n} \in \mathbb{C}^d \right\}.$

This notion was first introduced in . In  it was shown that the maximal numerical range of an operator has a non-empty intersection with the boundary of its numerical range if and only if the operator is normaloid. More properties about maximal numerical range for specific class of matrices we can find in .

1. Joseph Stampfli, 1970. The norm of a derivation. Pacific journal of mathematics, 33, Mathematical Sciences Publishers, pp.737–747.
2. Ilya M Spitkovsky, 2018. A note on the maximal numerical range. arXiv preprint arXiv:1803.10516.
3. Ali Hamed, Ilya Spitkovsky, 2018. On the maximal numerical range of some matrices. The Electronic Journal of Linear Algebra, 34, pp.288–303. 