Fredholm properties of the difference of orthogonal projections in a Hilbert space (Q1780976)

Let \(P\) and \(Q\) be orthogonal projections on an infinite-dimensional complex Hilbert space. The main result of this paper gives four conditions equivalent to \(P-Q\) being a Fredholm operator; for example, the essential norm (i.e., the distance to the compact operators) of \(P+Q-I\) being strictly less than \(1\), or \(P+Q\) and \(I-PQ\) being Fredholm operators. Moreover, explicit formulas for the Moore--Penrose inverse and for a Fredholm inverse of \(P-Q\) are given. Here \(S\) is a Fredholm inverse of \(T\) when \(I-ST\) and \(I-TS\) are finite rank operators.