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

Let N be a maximal atomic nest on Hilbert space H Alg(N) denote the associated nest algebra. We prove that a weakly continuous and surjective linear map (phi) : Alg(N) - Alg(N) preserves the closure of numerical range if and only if there exists a unitary operator U (element of) B(H) such that phi(T) = UTU* for every T (element of) Alg(N) or (phi)(T) = UT^(tr)U* for every T (element of) Alg(N), where T^(tr) denotes the transpose of T relative to an arbitrary but fixed base of H. As applications, we get the characterizations of the numerical range or numerical radius preservers on Alg(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.