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numerical-range:generalizations:maximal-numerical-range

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 numerical-range:generalizations:maximal-numerical-range [2018/10/08 08:20]plewandowska numerical-range:generalizations:maximal-numerical-range [2018/10/08 08:21] (current)plewandowska [Definition] Both sides previous revision Previous revision 2018/10/08 08:21 plewandowska [Definition] 2018/10/08 08:20 plewandowska 2018/05/21 17:29 lpawela 2018/05/20 13:26 lpawela created 2018/10/08 08:21 plewandowska [Definition] 2018/10/08 08:20 plewandowska 2018/05/21 17:29 lpawela 2018/05/20 13:26 lpawela created Line 4: Line 4: Let $A$ be an $d \times d$ matrix. The maximal numerical range of $A$ is the set: Let $A$ be an $d \times d$ matrix. The maximal numerical range of $A$ is the set: $$$- W_0(A) = \left\{ z \in \mathbb{C}: \bra{x_n}A\ket{x_n} \to z, \|A\ket{x_n} \| \to \|A\|, \braket{x_n}{x_n} = 1, \ket{x_n} \in \mathbb{C}^N \right\}. + W_0(A) = \left\{ z \in \mathbb{C}: \bra{x_n}A\ket{x_n} \to z, \|A\ket{x_n} \| \to \|A\|, \braket{x_n}{x_n} = 1, \ket{x_n} \in \mathbb{C}^d \right\}.$$$ This notion was first introduced in [( :​stampfli1970norm )]. In [( :​spitkovsky2018note )] 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. This notion was first introduced in [( :​stampfli1970norm )]. In [( :​spitkovsky2018note )] 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.