Unital Positive Linear Maps Research Articles

Two-moment characterization of spectral measures on the real line

Abstract In Kiukas, Lahti, and Ylinen (2006, Journal of Mathematical Physics 47, 072104), the authors asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in Pietrzycki and Stochel (2021, Journal of Functional Analysis 280, 109001). Let T be a self-adjoint operator, and let F be a Borel semispectral measure on the real line with compact support. For which positive integers $p< q$ do the equalities $T^k =\int _{\mathbb {R}} x^k F(\mathrm {d\hspace {.1ex}} x)$ , $k=p, q$ , imply that F is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if $p$ is odd and $q$ is even, and negative otherwise. The case $(p,q)=(1,2)$ closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck, as well as Kiukas, Lahti, and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on $C^*$ -algebras is also provided.

New generalized inequalities using arbitrary operator means and their duals

In this article, we present some operator inequalities via arbitrary operator means and unital positive linear maps. For instance, we show that if $A,B \in {\mathbb B}({\mathscr H}) $ are two positive invertible operators such that $ 0 < m \leq A,B \leq M $ and $\sigma$ is an arbitrary operator mean, then \begin{align*} \Phi^{p}(A\sigma B) \leq K^{p}(h) \Phi^{p}(B\sigma^{\perp} A), \end{align*} where $\sigma^{\perp}$ is dual $\sigma$, $p\geq0$ and $K(h)=\frac{(M+m)^{2}}{4 Mm}$ is the classical Kantorovich constant. We also generalize the above inequality for two arbitrary means $\sigma_{1},\sigma_{2}$ which lie between $\sigma$ and $\sigma^{\perp}$.

Lipschitz isomorphisms of compact quantum metric spaces

A compact quantum metric space is a complete order unit space endowed with a Lip-norm. We study morphisms and isomorphisms between compact quantum metric spaces. We give two sufficient conditions for a unital positive linear map of a compact quantum metri

Asymmetric Choi–Davis inequalities

ABSTRACT Let Φ be a unital positive linear map and let A be a positive invertible operator. We prove that there exist partial isometries U and V such that and hold under some mild operator convex conditions and some positive numbers r. Further, we show that if is operator concave, then In addition, we give some counterparts to the asymmetric Choi–Davis inequality and asymmetric Kadison inequality. Our results extend some inequalities due to Bourin–Ricard and Furuta.

Estimates of Ando–Hiai-type inequalities on operator power means

ABSTRACT We aim to give Ando–Hiai-type ratio inequalities for an operator power mean and its reverse ones which represent a relation between and for r>0 under k-tuple of positive invertible operators and . As applications, we consider complementary inequalities related to the information monotonicity property via any unital positive linear map. In particular, we state some results for the Karcher means by .

VARIANTS OF ANDO–HIAI TYPE INEQUALITIES FOR DEFORMED MEANS AND APPLICATIONS

AbstractFor an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

Improvements of operator reverse AM-GM inequality involving positive linear maps

In this paper, we shall present some reverse arithmetic-geometric mean operator inequalities for unital positive linear maps. These inequalities improve some corresponding results due to Xue (J. Inequal. Appl. 2017:283, 2017).

Lipschitz isometries of compact quantum metric spaces

A compact quantum metric space is a complete order unit space endowed with a Lip-norm. We investigate the category morphism between compact quantum metric spaces. We show that a unital positive linear map between compact quantum metric spaces with lower semicontinuous Lip-norms is Lipschitz if and only if it preserves the Lipschitz elements. We characterize when a unital positive linear map on a compact quantum metric space with lower semicontinuous Lip-norm is a Lipschitz isometry.

Variants of Ando–Hiai inequality for operator power means

It is known that for every and every k-tuple of positive invertible operators , the Ando–Hiai type inequality for operator power means holds, where is the operator norm and is the operator power mean. However it is not known any relation between and under the Löwner partial order. In this paper, we present some Ando–Hiai type inequalities for operator power means, which give a relation between and for every unital positive linear map Φ. In addition, we obtain a difference counterpart to the information monotonicity.

Several Inequalities for Khatri-Rao Products of Hilbert Space Operators

We establish several inequalities for Khatri-Rao products of Hilbert space operators, involving ordinary products, ordinary powers, ordinary inverses, and Moore-Penrose inverses. Kantorovich type inequalities concerning Khatri-Rao products are also investigated. Our results generalize some matrix inequalities in the literature. In our case, we must impose some mild conditions on operators such as the closeness of their ranges. Furthermore, we develop new operator inequalities by using block partitioning technique and unital positive linear maps.

Further generalizations of some operator inequalities involving positive linear map

We obtain a generalized conclusion based on an ?-geometric mean inequality. The conclusion is presented as follows: If m1,M1,m2,M2 are positive real numbers, 0 < m1 ? A ? M1 and 0 < m2 ? B ? M2 for m1 < M1 and m2 < M2, then for every unital positive linear map ? and ? ? (0,1], the operator inequality below holds: (?(?)#??(B))p ? 1/16 {(M1+m1)2((M1+m1)-1(M2+m2))2?)/(m2M2)?(m1M1)1- ?}p ?p(A#?B), p ? 2. Likewise, we give a second powering of the Diaz-Metcalf type inequality. Finally, we present p-th powering of some reversed inequalities for n operators related to Karcher mean and power mean involving positive linear maps.

Grüss type inequalities for positive linear maps on C∗-algebras

Let and be two unital -algebras and let for . We prove that if is a unital positive linear map, thenfor all and -pagination In addition, we show that if is a noncommutative probability space and is a density operator, thenandfor every and . Our results generalize the corresponding results for matrices to operators on spaces of arbitrary dimension.

Reverse Jensen-Mercer Type Operator Inequalities

Let $A$ be a selfadjoint operator on a Hilbert space $\mathcal{H}$ with spectrum in an interval $[a,b]$ and $\phi:B(\mathcal{H})\rightarrow B(\mathcal{K})$ be a unital positive linear map, where $\mathcal{K}$ is also a Hilbert space. Let $m,M\in J$ with $m

Khatri-Rao Products for Operator Matrices Acting on the Direct Sum of Hilbert Spaces

We introduce the notion of Khatri-Rao product for operator matrices acting on the direct sum of Hilbert spaces. This notion generalizes the tensor product and Hadamard product of operators and the Khatri-Rao product of matrices. We investigate algebraic properties, positivity, and monotonicity of the Khatri-Rao product. Moreover, there is a unital positive linear map taking Tracy-Singh products to Khatri-Rao products via an isometry.

On some inequalities with matrix means

In this note, we generalize some inequalities with matrix means due to Hoa et al. [RIMS Kokyuroku. 1893;2013:67–71]. Let and two arbitrary means between harmonic and arithmetic means. Then for every unital positive linear map and ,and

On matrix inequalities between the power means: Counterexamples

We prove that the known sufficient conditions on the real parameters (p,q) for which the matrix power mean inequality ((Ap+Bp)/2)1/p⩽((Aq+Bq)/2)1/q holds for every pair of matrices A,B>0 are indeed best possible. The proof proceeds by constructing 2×2 counterexamples. The best possible conditions on (p,q) for which Φ(Ap)1/p⩽Φ(Aq)1/q holds for every unital positive linear map Φ and A>0 are also clarified.

Squaring a reverse AM-GM inequality

Let $A , B$ be positive operators on a Hilbert space with $0 < m \le A, B \le M$. Then for every unital positive linear map $\varPhi $, $$ \varPhi ^2((A+B)/2)\le K^2(h)\varPhi ^{2}(A\mathbin {\sharp } B), $$ and $$ \varPhi ^2((A+B)/2)\le K^2(h)(\v

Bounds for an operator concave function

Letandbe unital positive linear maps satisfying some conditions with respect to positive scalarsand �. It is shown that if a real valued function f is operator concave on an interval J, then � (f(�(A)) �(f(A))) � f(�(A)) �(f(A)) � �(f(�(A)) �(f(A))) for every selfadjoint operator A with spectrum �(A) � J. Moreover, an external version of estimates above is presented.

An operator inequality and its consequences

Let f be a continuous convex function on an interval J, let A, B, C, D be self-adjoint operators acting on a Hilbert space with spectra contained in J such that A+D=B+C and A⩽m⩽B,C⩽M⩽D for two real numbers m<M, and let Φ be a unital positive linear map on B(H). We prove the inequalityf(Φ(B))+f(Φ(C))⩽Φ(f(A))+Φ(f(D))and apply it to obtain several inequalities such as the Jensen–Mercer operator inequality and the Petrović operator inequality.

Extensions of Kadison’s inequality on positive linear maps

Let U be a unital C ∗ -algebra, B ( H ) the algebra of all bounded linear operators on a Hilbert space H , and P [ U , B ( H ) ] the set of all positive linear maps from U to B ( H ) . The well-known Kadison’s inequality on unital positive linear maps is said that, if Φ ∈ P [ U , B ( H ) ] and Φ is unital, then Φ ( A 2 ) ≥ Φ ( A ) 2 for each Hermitian A . This paper is to consider the extensions of Kadison’s inequality, some inequalities for unital Φ ∈ P [ U , B ( H ) ] are obtained which generalize Furuta’s result, and a complement to a result of Bourin and Ricard is provided.