Fixed product preserving mappings on Banach algebras (Q2674343)

Let \(\mathcal{A}\) and \(\mathcal{B}\) be two complex unital Banach algebras and let \(\Phi: \mathcal{A} \to \mathcal{B}\) be a bijective linear map. Given \(c \in \mathcal{A}\) and \(d \in \mathcal{B}\), the author investigates the following general property \[ ab=c \Rightarrow \phi a \; \phi b=d. \] Typical examples are scalar multiples of isomorphisms. The case where \(\mathcal{A}= \mathcal{B}=M_n(\mathbb{C})\) was treated in [\textit{L. Catalano} and \textit{H. Julius}, J. Algebra 575, 220--232 (2021; Zbl 1475.15036); \textit{C. Costara}, J. Algebra 587, 336--343 (2021; Zbl 1478.15004)]. In the present paper the author provides a complete description of the following cases: \begin{itemize} \item[(1)] \(\mathcal{A}= \mathcal{B}(X)\), the algebra formed by all bounded operators on an infinite dimensional complex Banach space \(X\), \(\mathcal{B}\) is a prime algebra, and the operator \(c\) has finite rank. \item[(2)] \(d\) is invertible. \end{itemize} Moreover, he investigates the relation between the elements \(c\) and \(d\) in case \(\Phi\) exists.