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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 [1]. In [2] 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.

1. Joseph Stampfli, 1970. The norm of a derivation. Pacific journal of mathematics, 33, Mathematical Sciences Publishers, pp.737–747.