Characterization of Numerical Radius Preserving Isomorphisms in Norm-Attainable Classes

In functional analysis, understanding when an isomorphism preserves numerical radius in norm-attainable classes remains interesting. This paper examines conditions under which isomorphisms preserve the numerical radius in norm-attainable classes. We develop and prove conditions that provide insights into the behavior of isomorphisms concerning numerical radius preservation. The results show that if an isomorphism Φ preserves the numerical radius for all self-adjoint operators, then Φ is either a unitary operator or the negative of a unitary operator. Moreover, if Φ is an isomorphism in NA(H), then Φ is an orthogonal isomorphism.