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

2020 ◽  
pp. 1-18 ◽  
Author(s):  
MOHSEN KIAN ◽  
MOHAMMAD SAL MOSLEHIAN ◽  
YUKI SEO
Keyword(s):  
Hilbert Space ◽  
Norm Inequality ◽  
Power Mean ◽  
Variable Operator ◽  
Linear Map ◽  
Power Means ◽  
Operator Mean ◽  
Positive Linear Map ◽  
Euclidean Mean

Abstract For 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.

Reverse Jensen-Mercer Type Operator Inequalities

2016 ◽  
Vol 31 ◽  
pp. 87-99 ◽  
Author(s):  
Ehsan Anjidani ◽  
Mohammad Reza Changalvaiy
Keyword(s):  
Hilbert Space ◽  
Selfadjoint Operator ◽  
Type Operator ◽  
Operator Inequalities ◽  
Linear Map ◽  
Positive Linear Map

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

scholarly journals Further generalizations of some operator inequalities involving positive linear map

Filomat ◽  
2017 ◽  
Vol 31 (8) ◽  
pp. 2355-2364 ◽  
Author(s):  
Changsen Yang ◽  
Chaojun Yang
Keyword(s):  
Geometric Mean ◽  
Type Inequality ◽  
Operator Inequality ◽  
Power Mean ◽  
Positive Real ◽  
Operator Inequalities ◽  
Linear Map ◽  
Linear Maps ◽  
Positive Linear Map ◽  
Karcher Mean

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.

scholarly journals Conditional Expectations for Unbounded Operator Algebras

2007 ◽  
Vol 2007 ◽  
pp. 1-22 ◽  
Author(s):  
Atsushi Inoue ◽  
Hidekazu Ogi ◽  
Mayumi Takakura
Keyword(s):  
Hilbert Space ◽  
Conditional Expectation ◽  
Unbounded Operator ◽  
Operator Algebras ◽  
Linear Map ◽  
Conditional Expectations ◽  
Positive Linear Map

Two conditional expectations in unbounded operator algebras (O∗-algebras) are discussed. One is a vector conditional expectation defined by a linear map of anO∗-algebra into the Hilbert space on which theO∗-algebra acts. This has the usual properties of conditional expectations. This was defined by Gudder and Hudson. Another is an unbounded conditional expectation which is a positive linear mapℰof anO∗-algebraℳonto a givenO∗-subalgebra𝒩ofℳ. Here the domainD(ℰ)ofℰdoes not equal toℳin general, and so such a conditional expectation is called unbounded.

Weak Semiprojectivity in Purely Infinite Simple C*-Algebras

2007 ◽  
Vol 59 (2) ◽  
pp. 343-371 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Finite Subset ◽  
Finitely Generated ◽  
Linear Map ◽  
C Algebras ◽  
Direct Sum ◽  
Finitely Generated Groups ◽  
Positive Linear Map ◽  
Universal Coefficient Theorem

AbstractLet A be a separable amenable purely infinite simple C*-algebra which satisfies the Universal Coefficient Theorem. We prove that A is weakly semiprojective if and only if Ki(A) is a countable direct sum of finitely generated groups (i = 0, 1). Therefore, if A is such a C*-algebra, for any ε > 0 and any finite subset ℱ ⊂ A there exist δ > 0 and a finite subset ⊂ A satisfying the following: for any contractive positive linear map L : A → B (for any C*-algebra B) with ∥L(ab) – L(a)L(b)∥ < δ for a, b ∈ there exists a homomorphism h: A → B such that ∥h(a) – L(a)∥ < ε for a ∈ ℱ.

scholarly journals Optimal Inequalities for Power Means

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Yong-Min Li ◽  
Bo-Yong Long ◽  
Yu-Ming Chu ◽  
Wei-Ming Gong
Keyword(s):  
Power Mean ◽  
Power Means ◽  
Optimal Inequalities

We present the best possible power mean bounds for the productMpα(a,b)M-p1-α(a,b)for anyp>0,α∈(0,1), and alla,b>0witha≠b. Here,Mp(a,b)is thepth power mean of two positive numbersaandb.

Riesz spaces with the order-continuity property. I

1977 ◽  
Vol 81 (1) ◽  
pp. 31-42 ◽  
Author(s):  
D. H. Fremlin
Keyword(s):  
Riesz Space ◽  
Continuity Property ◽  
Sequential Order ◽  
Order Continuity ◽  
Linear Map ◽  
Riesz Spaces ◽  
Quotient Spaces ◽  
Positive Linear Map ◽  
Order Continuous

A Riesz space E has the (sequential) order-continuity property if every positive linear map from E to an Archimedean Riesz space is (sequentially) order-continuous. This is the case if and only if the canonical maps from E to its Archimedean quotient spaces are all (sequentially) order-continuous. I relate these properties to others that have been described elsewhere.

Riesz spaces with the order-continuity property. II

1978 ◽  
Vol 83 (2) ◽  
pp. 211-223 ◽  
Author(s):  
D. H. Fremlin
Keyword(s):  
Internal Structure ◽  
Riesz Space ◽  
Continuity Property ◽  
Order Continuity ◽  
Linear Map ◽  
Riesz Spaces ◽  
Positive Linear Map ◽  
Order Continuous

I continue to investigate Riesz spaces E with the property that every positive linear map from E to an Archimedean Riesz space is sequentially order-continuous. In order to give a criterion for the product of such spaces to be another, we are forced to investigate their internal structure, and to develop an ordinal hierarchy of such spaces.

scholarly journals Remarks on an operator Wielandt Inequality

2015 ◽  
Vol 30 ◽  
pp. 577-584
Author(s):  
Pingping Zhang
Keyword(s):  
Hilbert Space ◽  
Recent Result ◽  
Positive Operator ◽  
Upper Bounds ◽  
Linear Map ◽  
Wielandt Inequality

Let A be a positive operator on a Hilbert space H with 0 < m ≤ A ≤ M, and let X and Y be isometries on H such that X*Y = 0, p > 0, and Φ be a 2-positive unital linear map. Define Γ = (Φ(X*AY )Φ(Y*AY )^(−1)Φ(Y*AX)^p Φ(X*AX)^(−p). Several upper bounds for (1/2) |Γ + Γ*| are established. These bounds complement a recent result on the operator Wielandt inequality.

scholarly journals Operator Jensen's Inequality for Operator Superquadratic Functions

2019 ◽  
Author(s):  
Mohammad W. Alomari
Keyword(s):  
Hilbert Space ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Linear Map ◽  
Superquadratic Function ◽  
Hilbert Space Operators ◽  
Superquadratic Functions

In this work, an operator superquadratic function (in operator sense) for positive Hilbert space operators is defined. Several examples with some important properties together with some observations which are related to the operator convexity are pointed out. Equivalent statements of a non-commutative version of Jensen's inequality for operator superquadratic function are established. A generalization of the main result to any positive unital linear map is also provided.

Inequalities for relative operator entropies

2014 ◽  
Vol 27 ◽  
Author(s):  
Pawel Kluza ◽  
Marek Niezgoda
Keyword(s):  
Linear Algebra ◽  
Operator Inequalities ◽  
Linear Map ◽  
Positive Linear Map ◽  
Kantorovich Constant ◽  
Reverse Inequalities

In this paper, operator inequalities are provided for operator entropies transformed by a strictly positive linear map. Some results by Furuichi et al. [S. Furuichi, K. Yanagi, and K. Kuriyama. A note on operator inequalities of Tsallis relative operator entropy. Linear Algebra Appl., 407:19–31, 2005.], Furuta [T. Furuta. Two reverse inequalities associated with Tsallis relative operator entropy via generalized Kantorovich constant and their applications. Linear Algebra Appl., 412:526–537, 2006.], and Zou [L. Zou. Operator inequalities associated with Tsallis relative operator entropy. Math. Inequal. Appl., 18:401–406, 2015.] are extended. In particular, the obtained inequalities are specified for relative operator entropy and Tsallis relative operator entropy. In addition, some bounds for generalized relative operator entropy are established.

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