On Fixed Points Of Completely Positive Maps And Reverse-order Laws For Operator Products

Operator equation is one of the hotest topics in operator theory and operator algebra. Some problems in physics or optimization theory can be settled by solving linear or nonlinear operator equations. Many people has paied close attation to some certain operator eqations. Two kinds of operator equations are investigated in this thesis, which are operator eqations for fixed points associated to completely positive maps and Moore-Penrose equations.Completely positive map is an important research object in operator algebra, which has been studied by many scholars. Especially recent years, it has a great developtment since a quantum channel can be represented by a trace preserving com-pletely positive map. Here, we mainly study the completely positive map decided by an operator sequence. Let H and K be Hilbert spaces, B(H), B{K,H) denote the set of all bounded linear operators on H and from H into K, respectively. Denote by J a finite or countable index set. A sequence A={Ak}k∈J of bounded operators on H is called a row contraction if∑k∈J AkAk*≤IH, where IH is the identity operator on H and the series is convergent in the strong operator topology when J is an infinite set. To each row contraction A={Ak}k∈J one can associate a normal completely positive mapping ΦA:B(H)�'B(H) defined by ΦA(X)＝∑k∈JAkXAk*,(?)X∈B(H). In this caseΦA is called a quantum operation. At the same time, if∑k∈J Ak*Ak≤I holds,then ΦA+(X)=＝∑k∈JAk*XAk,(?)X∈B(H). is well defined, which is called the dual operation of ΦA. If the operator X∈B(H) satisfies operator equation ΦA(X)=X, then X is called the fixed point of ΦA. Denote by B{H)ΦA the fixed point set of ΦA.Firstly, by use of block-operator matrix technique and dilation theory, we discuss some characterization of commutative operator sequence. Based on above results, fixed points of completely positive map and dual operation decided by commutative row contractions are studied. At last, we research reverse-order laws for{1,3}-and {1,2,3}-inverse according to certain space decompositions. There are four chapters in this thesis, and the main contents are as follows.In the first chapter, we introduce some research status and background about the fixed point of quantum operation and reverse-order laws. And also introduce some notations and lemmas.In the second chapter, some characterization of commutative operator sequence are studied. Firstly, by use of normal dilation of commutative row contraction, some sufficient conditions for unital commutative row contraction to be normal are given. And then B(H)ΦA A’ under this conditions. Moreover, lifting of commutative operator sequence are considered and unital commutative and pure row contractions are characterized. At last, we discuss the commutant of normal commutative operator sequence A and have the equality (A·A)’=A’+Iw(A,-A), where Iw(A,-A)={X∈B(H)|AkX=-XAk,(?)k}.In the third chapter, we investigate fixed points as well as completely disturbed point of quantum and dual quantum operation. At first, we consider lifting of fixed point of Φ A,B and give a sufficient condition of B(K,H)ΦA,B={0}, which will be useful. Next, by studying some sufficient conditions for the operator sequence ΦAj(I) conver-gent to a projection, we give a characterization of fixed points set B(H)ΦA. Besides, we give a equivalent condition for operator sequence ΦAj(I) converges to a projection in strong operator topology. And then the fixed point set of quantum channel are characterized. Moreover, completely disturbed points of quantum operation are re-searched and some sufficient condition are discussed for the equation ΦA(X)=I-X has only one solution1/2I. At last, we study the relationship between the fixed point of quantum operation and dual quantum operation.In the fourth chapter, we investigate reverse-order laws for generalize inverses of the two-operator product by making full use of block-operator matrix technique. The necessary and sufficient conditions for BθAθ=(AB)θ are presented when the range of A, B, AB are closed, where θ∈{{1,3},{1,4},{1,2,3},{1,2,4}}. Moreover, the reverse-order law for Moore-Penrose inverse is also considered and the equivalent condition for (AB)+=B+A+is given.