Special Classes of Bounded Operators
Philippe Blanchard, Erwin Brüning · Birkhäuser Boston eBooks · 2003
Let e be a unit vector in a Hilbert space H, over the field K with inner product 〈•, •〉. Define P e : H → H by P e x = 〈e, x〉e for all x ∈ H. Evidently, P e is a bounded linear operator with null space N(P e ) = {e}⊥ and range ran P e = Ke. In addition P e satisfies and which is also elementary to prove. The operator P e is the simplest example of the class of projection operators or projectors to be studied in this section.