Linear mappings preserving the diameter of the numerical range
Let\(\mathcal{\;H}\) be a complex Hilbert space and \(\mathcal{B(H)}\) the algebra of all bounded linear operators on \(\mathcal{H}\). For \(A\in \mathcal{B(H)}\), the numerical range of \(A\) is defined by \(W(A)=\left\{ \left\langle Ax,x\right\rangle :x\in \mathcal{H}\text{ and }\left\| x\right\| =1\right\} \). The diameter of \(W(A)\) is defined as \[ d(W(A))=\sup \left\{ \left| \lambda -\mu \right| :\lambda ,\mu \in W(A)\right\} . \] We say that a map \(\Psi :\mathcal{B(H)\rightarrow B(H)}\) preserves the diameter if \[ d(\Psi (W(A)))=d(W(A)) \] for all \(A\in \mathcal{B(H)}\). In the paper under review, authors study linear maps on \(\mathcal{B(H)}\) that preserve the diameter. They prove the following result. Theorem. Let \(\Psi :\mathcal{B(H)}\rightarrow\mathcal{B(H)}\) be a surjective linear map that preserves the diameter. Then there exist \(\lambda \in \mathbb{C}\) with \(\left| \lambda \right| =1\), a unitary operator \(U\in \mathcal{B(H)}\), and a linear functional \(\Lambda :\mathcal{B(H)}\rightarrow \mathbb{C}\) satisfying \(\Lambda (I)+\lambda \neq 0\) such that \(\Psi \) is either of the form \[ \Psi (A)=\lambda UAU^{\ast }+\Lambda (A)I\quad \text{for every }A\in \mathcal{B(H)} \] or of the form \[ \Psi (A)=\lambda UA^{t}U^{\ast }+\Lambda (A)I\quad \text{for every }A\in \mathcal{B(H)}, \] where \(I\) is the identity operator on \(\mathcal{H}\) and \(A^{t}\) denotes the transpose of \(A\) taken with respect to an orthonormal basis fixed in advance. The proof of this theorem is divided into two main parts. Let \(\left\langle I\right\rangle \) denote the linear span of \(I\). Consider the dual space \(\left( \mathcal{B(H)}/\left\langle I\right\rangle \right) ^{\ast }\) of the quotient space \(\mathcal{B(H)}/\left\langle I\right\rangle \). In the first part of the proof, authors give a characterization of the extreme functionals, i.e., the extreme points of the closed unit ball of \(\left( \mathcal{ B(H)}/\left\langle I\right\rangle \right) ^{\ast }\). They use this characterization in the second part of the proof where another main tool is used -- namely, a result of \textit{L. Molnár} and \textit{M. Barczy} [J. Funct. Anal. 205, No. 2, 380--400 (2003; Zbl 1047.47029)] which describes the form of all bijective linear maps that preserve the diameter on the set of all self-adjoint operators in \(\mathcal{B(H)}\).
scientific article; zbMATH DE number 6756330
scientific article