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Maximal numerical range


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

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.
2. Ilya M Spitkovsky, 2018. A note on the maximal numerical range. arXiv preprint arXiv:1803.10516.
numerical-range/generalizations/maximal-numerical-range.1538986815.txt.gz · Last modified: 2018/10/08 08:20 by plewandowska