Isometries of a generalized numerical radius

Abstract For 0 q 1 , the q -numerical range is defined on the algebra M n of all n × n complex matrices by W q ( A ) = { x ∗ Ay : x , y ∈ C n , ∥ x ∥ = ∥ y ∥ = 1 , 〈 y , x 〉 = q } . The q -numerical radius is defined by r q ( A ) = max { | μ | : μ ∈ W q ( A ) } . We characterize isometries of the metric space ( M n , r q ) , i.e., the maps φ : M n → M n that satisfy r q ( A - B ) = r q ( φ ( A ) - φ ( B ) ) . We also characterize maps on M n that preserves the q -numerical range. The maps are not assumed to be linear.