New Results on Idempotent Operators in Hilbert Spaces

Abstract

This paper provides a new proof of the operator norm identity ∥Q∥ = ∥I −Q∥, where Q is abounded idempotent operator on a complex Hilbert space, and I is the identity operator. Wealso derive explicit lower and upper bounds for the distance from an arbitrary idempotentoperator to the set of orthogonal projections. Our approach simplifies existing proofs.