A unified formulation is given to various generalizations of the classical numerical range including the $c$-numerical range, congruence numerical range, $q$-numerical range and von Neumann range. Attention is given to those cases having connections with classical simple real Lie algebras. Convexity and inclusion relation involving those generalized numerical ranges are investigated. The underlying geometry is emphasized.

Keywords:

numerical range, convexity, inclusion relation numerical range, convexity, inclusion relation

MSC Classifications:

15A60, 17B20 show english descriptions Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]
Simple, semisimple, reductive (super)algebras 15A60 - Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]
17B20 - Simple, semisimple, reductive (super)algebras

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