Maps preserving local spectral subspaces of generalised product of operators

Abstract

Let \(\mathcal {B}(X)\) be the algebra of all bounded linear operators on a complex Banach space X. For \(A\in \mathcal {B}(X)\) and \(\lambda \in \mathbb {C}\), let \(X_{A}(\{\lambda \})\) be the local spectral subspace of A associated with \(\{\lambda \}\). For an integer \(k\ge 2\), let \((i_1,\ldots ,i_m)\) be a finite sequence with terms chosen from \(\{1,\ldots ,k\}\), such that \(\{i_1,\ldots ,i_m\}=\{1\ldots k\}\) and at least one of the terms in \(\{i_1,\ldots ,i_m\}\) appears exactly once. The generalized product of k operators \(A_1\cdots A_k \in \mathcal {B}(X)\) is defined by $$\begin{aligned} A_1*A_2*\cdots *A_k=A_{i_{1}}A_{i_{2}}\cdots A_{i_{m}}, \end{aligned}$$ and includes the usual product and the triple product. We characterise the form of surjective maps from \(\mathcal {B}(X)\) into itself satisfying $$\begin{aligned} X_{\phi (A_{1})*\cdots *\phi (A_{k})}(\{\lambda \})=X_{A_{1}*\cdots *A_{k}}(\{\lambda \}), \end{aligned}$$ for all \(A_1,\ldots ,A_k \in \mathcal {B}(X)\) and \(\lambda \in \mathbb {C}\).