Linear Maps Preserving the Closure of Numerical Range on Nest Algebras with Maximal Atomic Nest

Abstract

Let \( \mathcal{N} \) be a maximal atomic nest on Hilbert space H and \( \textrm{Alg}\mathcal{N} \)denote the associated nest algebra. We prove that a weakly continuous andsurjective linear map \( \Phi\,:\,\textrm{Alg}\mathcal{N}\longrightarrow\textrm{Alg}\mathcal{N} \) preserves the closure of numericalrange if and only if there exists a unitary operator \( U\in\mathcal{B}(H) \) such that\( Phi(T) = UTU^{*} \) for every \( T\in\textrm{Alg}\mathcal{N} \) or \( \Phi(T) = UT^{tr}U^{*} \) for every \( T\in\textrm{Alg}\mathcal{N} \),where \( T^{tr} \) denotes the transpose of T relative to an arbitrary but fixed baseof H. As applications, we get the characterizations of the numerical rangeor numerical radius preservers on \( \textrm{Alg}\mathcal{N} \). The surjective linear maps on the diagonal algebras of atomic nest algebras preserving the closure of numerical range or preserving the numerical range (radius) are also characterized.