On centralizers of reflexive algebras (Q1759602)

Let \(\mathcal{L}\) be a subspace lattice on a complex Banach space \(X\) and \(\delta:alg\mathcal{L} \to B(X)\) a linear mapping such that for every \(A \in alg\mathcal{L}\), either \(2\delta(A^2)=\delta(A)A + A\delta(A)\) or \(\delta(A^3) = A\delta(A)A\). In the paper, the authors show that, if \(\vee\{L : L \in \mathcal{J}(\mathcal{L})\}=X\) or \(\wedge\{L_-: L \in \mathcal{J}(\mathcal{L})\}=(0)\) and \(X\) is reflexive, then \(\delta\) is a centralizer. Moreover, the authors prove that, when \(\mathcal{L}\) is a CSL and \(\delta\) is a linear mapping from \(alg\mathcal{L}\) into itself, then \(\delta\) is a centralizer.