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

Abstract

Let N be a maximal atomic nest on Hilbert space H and AlgN denote the associated nest algebra. We prove that a weakly continuous and surjective linear map Φ : AlgN →AlgN preserves the closure of numerical range if and only if there exists a unitary operator U ∈ B(H) such that Φ(T ) = UT U * for every T ∈AlgN or Φ(T ) = UT tr U * for every T ∈AlgN , 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 AlgN . 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. Mathematics Subject Classification (2000). Primary 47B48; Secondary 47A12, 47L35.

Linear Maps Preserving Numerical Radius on Nest Algebras

“…In [2,3], Chan proved that if A is a unital C * -algebra then a surjective numerical radius isometry of A is a Jordan isomorphism multiplied by a fixed unitary element in the center of A. This is also true for weakly continuous, surjective numerical radius isometries of atomic nest algebras [4,5,14]. Roughly speaking, nest algebras are reflexive algebras with ordered lattices.…”

“…Then P is an interval of N . (4). Suppose that P ∈ AlgN is a projection for which P (AlgN )P ⊥ (AlgN ) P = {0} and P ⊥ (AlgN )P (AlgN )P ⊥ = {0}.…”

“…Then there exists a scalar γ such that Proof. The proof is standard; the readers can refer [4,5,14].…”

“…The purpose of this paper is to characterize surjective numerical radius isometries of nest algebras. Unlike [4,5], we do not require isometries to be weakly continuous and do not require nest algebras to be atomic. Also, our approach is very different from that in [4,5].…”

Non-linear maps on self-adjoint operators preserving numerical radius and numerical range of Lie product

“…Let A be a bounded linear operator acting on a complex Hilbert space H. Recall that the numerical range of A is the set W (A) = { Ax, x | x ∈ H, x = 1}, and the numerical radius of A is w(A) = sup{|λ| | λ ∈ W (A)}. The problem of characterizing linear maps on matrices or operators that preserve numerical range or numerical radius has been studied by many authors, see for example [3,4,7,18] and the references therein. In recent years, interest in characterizing general (non-linear) preservers of numerical ranges or numerical radius has been growing ( [1,2,5,6,8,9,11,12,13,15,16,17,19]).…”