Research On Unitary Operator Functions And Symmetric Operators Based On Idempotent Operators

Let P be an idempotent operator on a Hilbert space H,it is easy to prove thatλP+I is invertible,when λ∈ C\{-1}.We denote two unitary operator functions Uλ and Vλ by Uλ:=(AP+I)|λP+I|-1 and Vλ:=(λP*+I)|λP*+I|-1,for λ∈ C\{-1}.And we denote byΓP={J:J=J*=J-1 and JPJ=I-P}and△P={J:J=J*=J-1 and JPJ=I-P*}.In this paper,we mainly study the properties and relations of the symmetries involving idempotents.The main contents are as follows:In Chapter 1,we introduce some basic concepts of symmetries,Krein space,J-projection,J-positive projection,J-negative projection,J-contractive projection,and some lemmes and theorems to be used in this paper.In Chapter 2,we mainly consider the set of all λ∈ C\{-1} which satisfy that Uλ and Vλ are symmetries or Uλ=Vλ.We firstly give the specific structures of |λP+I|-1 and |λP*+I|-1.Then the structures of Uλ and Vλ are established,respectively.For P ∈ B(H)Id\P(H),we get that Uλ=Vλ if and only if λ<-1 or|λ+1|=1.At the same time,Uλ is a symmetry if and only if λ<-1 or λ=0.Finally,we consider the limits(?)and(?)Vλ in the norm topology respectively.In Chapter 3,we mainly get that the specific structures of all symmetries J ∈ ΓP and J ∈ ΔP,respectively.Then we study the relations between J ∈ ΓP,J∈ΔP,and J when JP≥ 0.Moreover,we prove that J ∈ ΔP if and only if iJ(2P-I)|2PI|-1 ∈ΓP.