Unital Positive Linear Maps Research Articles

Bellman inequality for Hilbert space operators

We establish some operator versions of Bellman’s inequality. In particular, we prove that if Φ:B(H)→B(K) is a unital positive linear map, A,B∈B(H) are contractions, p>1 and 0⩽λ⩽1, thenΦ(IH-A∇λB)1/p⩾Φ(IH-A)1/p∇λ(IH-B)1/p.

Operator Aczél inequality

We establish several operator versions of the classical Aczél inequality. One of operator versions deals with the weighted operator geometric mean and another is related to the positive sesquilinear forms. Some applications including the unital positive linear maps on C * -algebras and the unitarily invariant norms on matrices are presented.

Around Choi inequalities for positive linear maps

We show an inequality for unital positive linear maps Φ interpolating Choi’s inequality (p=0) with a recent result of Bourin and Ricard (r=1):LetA∈A+,0⩽p⩽qandqq+p⩽r⩽2qq+p.Then|Φ(Ap)rΦ(Aq)r|⩽Φ(A(p+q)r).

Facial structures for unital positive linear maps in the two-dimensional matrix algebra

We completely determine the lattice structure for the faces of the convex set of all unital positive linear maps between the 2×2 matrix algebras.

Complementary inequalities to inequalities of Jensen and Ando based on the Mond–Pečarić method

Applying the Mond–Pečarić method to unital positive linear maps, we shall show several complementary inequalities to Jensen's inequalities on positive linear maps and consequently obtain complementary inequalities to Ando's inequalities associated with operator means. We shall apply them to obtain complementary estimates for the results by Ando, Aujla–Vasudeva and Fujii on Hadamard product and operator means.

Quasi-orders on the positive cone of aC∗

Motivated by an inequality of Choi for unital positive linear maps we introduce a quasi-order on the positive cone of a C ∗-algebra and present several intrinsic characterizations for it. We consider also another quasi-order based on an inequality of Kadison, and study interrelationship between those two quasi-orders.