Fixed Points Associated to Power of Normal Completely Positive Maps*

Let be a normal completely positive map with Kraus operators . An operator X is said to be a fixed point of , if . Let be the fixed points set of . In this paper, fixed points of are considered for , where means j-power of . We obtain that and for integral when A is self-adjoint and commutable. Moreover, holds under certain condition.

Completely Positive Definite Maps Over Topological-algebras

In this paper, we discuss completely positive definite maps over topological algebras. A Schwarz type inequality for n-positive definite maps, and the Stinespring representation theorem for completely positive definite maps over topological algebras are given.

A Classification of Completely Positive Maps

This paper concerns classifying completely positive maps between certain C*-algebras. Several invariants for classifying completely positive maps are constructed. It is proved that one of them is isomorphic to the Ext-group of C*-algebra extensions in special circumstances. Furthermore, this invariant induces a functor from C*-algebras to abelian groups which is split-exact.

Completely Positive Matrices Having Cyclic Graphs

We prove that a CP matrix A having cyclic graph has exactly two minimal rank 1 factorization if det M(A) > 0 and has exactly one minimal rank 1 factorization if detM(A) = 0.

Completely Positive Realizations of a Cycle

An n × n real matrix A is called doubly nounegative, if A is entrywise nonnegative and semidefmite positive as well. A is called completely positive if A can be factored as A=BBt,where B is some nonnegative n × m matrix. The smallest such number m is called the factorization index (or CP-rank) of A. This paper presents a criteria for a doubly nonnegative matrix realization of a cycle to be completely positive, which is strightforward and effective.

Completely positive tensors in the complex field

In this paper, we introduce the complex completely positive tensor, which has a symmetric complex decomposition with all real and imaginary parts of the decomposition vectors being non-negative. Some properties of the complex completely positive tensor are given. A semidefinite algorithm is also proposed for checking whether a complex tensor is complex completely positive or not. If a tensor is not complex completely positive, a certificate for it can be obtained;if it is complex completely positive, a complex completely positive decomposition can be obtained.

Factorization Index for Completely Positive Graphs

In this paper, we prove that the set of all factorization indices of a completely positive graph has no gaps. In other words, we give an affirmative answer to a question raised by N. Kogan and A. Berman [8] in the case of completely positive graphs.

COMPLETELY POSITIVE MATRICES OF ORDER FIVE

A real matrix A of order n is called doubly nonnegative (denoting A∈DPn) if it is non- negative entrywise and positive semidefinite as well. A is called completely positive (denoting A∈CPn) if there exist k nonnegative column vectors b1,b2, …,bk∈Rn for some nonnegative integer k such that A=b1b′1+…+bkb′k. The smallest such number k is called the factorization index of A and is denoted by φ(A). This paper gives an effective criterion for any doubly nonnegative matrix A of order 5 whose associated graph is isomorphic neither to K5(the complete graph) nor to K5-e (a subgraph of K5 obtained by cutting off an edge from it) to be completely positive.

Complete Positivity of Elementary Operators on C^＊—algebras

Let E(x) = be an elementary operator on a C-algebra A. We prove that if A is prime with soc(A) = 0, or there is a family of irreducible representation of A such that is faithful and (A) does not contain compact operator,or A has a faithful repre sentation π such that π(A)″with no central portions of type In for n>1 then E is positive if and only if E is completely positive.

Partially positive matrices

A real n × n symmetric matrix P is partially positive(PP) for a given index set I ? {1,..., n} if there exists a matrix V such that V(I, :) 0 and P = V VT. We give a characterization of PP-matrices. A semidefinite algorithm is presented for checking whether a matrix is partially positive or not. Its properties are studied. A PP-decomposition of a matrix can also be obtained if it is partially positive.

Analysis of Sparse Quasi-Newton Updates with Positive Definite Matrix Completion

Based on the idea of maximum determinant positive definite matrix completion,Yamashita(Math Prog 115(1):1–30,2008)proposed a new sparse quasi-Newton update,called MCQN,for unconstrained optimization problems with sparse Hessian structures.In exchange of the relaxation of the secant equation,the MCQN update avoids solving difficult subproblems and overcomes the ill-conditioning of approximate Hessian matrices.However,local and superlinear convergence results were only established for the MCQN update with the DFP method.In this paper,we extend the convergence result to the MCQN update with the whole Broyden’s convex family.Numerical results are also reported,which suggest some efficient ways of choosing the parameter in the MCQN update the Broyden’s family.

Pointwise Pseudo-Orbit Tracing Property and Chaos

In this article, we discuss the relationship between pointwise pseudo-orbit tracing property and chaotic properties such as topological mixing. When f has pointwise pseudo-orbit tracing property, we give some equal conditions of uniform positive entropy and completely positive entropy.