Research On Symmetric Operators In J-positive And J-squeezed Projections

Krein space is an important research field in functional analysis.In recent years,the study of its J-projections has attracted the attention of domestic and foreign scholars.They study J-projections by defining J as a bounded self-adjoint operator,conjugation operator and symmetry.Some conclusions of projections in Hilbert space are verified to be still valid in Krein space,and many special conclusions of J-projections are obtained.Based on the previous research,in this thesis,we mainly study a class of J-projections that the operator J is the symmetry.First we make the idempotent E and the symmetry J be decomposed under the space H=R(E)(?)R(E)⊥,and then chsaracterize the matrix form of symmetry J such that an idempotent E is J-(positive,contractive)projection,finally,we prove some existing conclusions of J-projections through these matrices,and obtain some new results.The main contents are as follows:In Chapter 1,we mainly introduce the symbols,concepts and theorems com-monly used in this paper.For example,Krein space,symmetry,J-(positive,nega-tive,contractive,expansive)projection,support projection and so on.In Chapter 2,we mainly study the minimal and maximal clement s of the set,of all symmetries J with JE≥0.That is min{J:JE)0,J=J*=J-1}=2P(E+E*)+-I.max{J:JE≥0,J=J*=J-1}=2P(E+E*)+-I+2PN(E+E*)-In Chapter 3,we mainly st udy the minimal and maximal elements of the set of all symmetries J with E*JE≤J.That is min{J:E*JE ≤J,J=J=J-1}=2P(E+E*)-I+2PN(E+E*),max{J:E*JE≤J,J=J*=J-1}=2P(E+E*)-I+2PN(E-E*)Moreover,some formulas between P(2I-E-E*)+(P(2I-E-E*)and P(E+E*)(P(E+E*)+)are established.P(2I-E-E*)+=P(E+E*)+PN(E+E*).P(2I-E-E*)+PN(2I-E-E*)=P(E+E*)+.