On linear maps preserving generalized invertibility and related properties (Q930972)

The general idea of linear preserver problems is to describe all maps \(\varphi: {\mathcal A}\to{\mathcal B}\) between Banach algebras which preserve some class of elements or some relation. The most famous is probably Kaplansky's problem which asks whether any surjective unital linear map \(\varphi\) between semi-simple Banach algebras which preserves invertibility has to be a Jordan homomorphism, i.e., satisfy \(\varphi(x^2)=\varphi(x)^2\) for all \(x\). Denote, respectively, by \({\mathcal B}(H)\), \({\mathcal F}(H)\) and \({\mathcal K}(H)\) the algebras of bounded, finite-rank and compact operators on the Hilbert space \(H\), and by \(\mathcal{C}(H):=\mathcal{B}(H)/\mathcal{K}(H)\) the Calkin algebra. Recall that an operator is called Fredholm if \(A+{\mathcal K}(H)\) is invertible in \({\mathcal C}(H)\) and semi-Fredholm if \(A+{\mathcal K}(H)\) is either left or right-invertible in \({\mathcal C}(H)\). For \(A\in {\mathcal B}(H)\), an element \(B\in{\mathcal B}(H)\) is called a generalized inverse of \(A\) if \(ABA=A\) and \(BAB=B\). In the paper under review, the authors consider the case of linear surjective maps \(\varphi:{\mathcal B}(H)\to {\mathcal B}(H)\) which preserve generalized invertibility. Their main theorem asserts that, if there is a semi-Fredholm operator \(U\) such that \(\varphi(U)\) is semi-Fredholm (this is the case, in particular, if \(\varphi\) is unital), then \(\varphi({\mathcal F}(H))\subset {\mathcal F}(H)\), \(\phi({\mathcal K}(H))\subset {\mathcal K}(H)\), and the induced map \(\widetilde{\varphi}: {\mathcal C}(H)\to {\mathcal C}(H)\) is of the form \(\widetilde{\varphi}(x)=a\psi(x)\), where \( a \in {\mathcal C}(H)\) is invertible and \(\psi\) is a Jordan homomorphism. This situation is slightly more general than the one treated in [\textit{M. Mbekhta} and \textit{P. Šemrl}, ``Linear maps preserving semi-Fredholm operators and generalized invertibility'', Linear Multilinear Algebra 57, No. 1, 55--64 (2009; Zbl 1201.47015) and \textit{M. Mbekhta, L. Rodman} and \textit{P. Šemrl}, ``Linear maps preserving generalized invertibility'', Integral Equations Oper. Theory 55, No. 1, 93--109 (2006; Zbl 1111.47031)], where generalized invertibility was preserved in both directions.