Unbounded or bounded idempotent operators in Hilbert space

Abstract

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 . © 2011 Elsevier Inc. All rights reserved.