Unbounded or bounded idempotent operators in Hilbert space

A densely defined, closed linear operator F in a Hilbert space is said to be idempotent if ran(F)⊂dom(F) and F·F=F. We show that such an idempotent operator is written as F=P(P+Q)-1/2·(P+Q)-1/2 where P and Q are the orthoprojections to ran(F) and ker(F), respectively. When F is bounded, this becomes F=P(P+Q)-1. Further we show that for any λ≠0 the operator P+λQ is invertible and F=P(P+λQ)-1. In addition to the known results we present several descriptions of the norm ‖F‖ in terms of ‖P+Q‖,‖(P+Q)-1‖ or ‖(P-Q)-1‖.