Core invertibility of operators from the algebra generated by two orthogonal projections

A Hilbert space operator A is said to be core invertible if it has an inner inverse whose range coincides with the range of A and whose null space coincides with the null space of the adjoint of A. This notion was introduced by Baksalary, Trenkler, Rakić, Dinčić, and Djordjević in the last decade, who also proved that core invertibility is equivalent to group invertibility and that the core and group inverses coincide if and only if A is a so-called EP operator. The present paper contains criteria for core invertibility and for the EP property as well as formulas for the core inverse for operators in the von Neumann algebra generated by two orthogonal projections.