Linear Maps Preserving Numerical Radius on Nest Algebras

Abstract

A linear map φ of operator algebras is said to preserve numerical radius ( or to be a numerical radius isometry) if w(φ(A)) = w(A) for all A in its domain algebra, where w(A) stands for the numerical radius of A. In this paper, we prove that a surjective linear map φ of the nest algebra AlgN onto itself preserves numerical radius if and only if there exist a unitary U and a complex number ξ of modulus one such that φ(A) = ξU AU * for all A ∈ AlgN , or there exist a unitary U , a conjugation J and a complex number ξ of modulus one such that φ(A) = ξU JA * JU * for all A ∈ AlgN .Mathematics Subject Classification (2010). Primary 47L35, 47B48; Secondary 47A12.