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 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.

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.

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.

scholarly journals An Operator Extension of Čebyšev Inequality

2017 ◽  
Vol 25 (2) ◽  
pp. 135-147
Author(s):  
Hamid Reza Moradi ◽  
Mohsen Erfanian Omidvar ◽  
Silvestru Sever Dragomir
Keyword(s):  
Compact Operator ◽  
Operator Inequalities ◽  
Linear Map ◽  
Positive Linear Map ◽  
Čebyšev Inequality ◽  
Operator Extension

Abstract Some operator inequalities for synchronous functions that are related to the čebyšev inequality are given. Among other inequalities for synchronous functions it is shown that ∥ø(f(A)g(A)) - ø(f(A))ø(g(A))∥ ≤ max{║ø(f2(A)) - ø2(f(A))║, ║ø)G2(A)) - ø2(g(A))║} where A is a self-adjoint and compact operator on B(ℋ ), f, g ∈ C (sp (A)) continuous and non-negative functions and ø: B(ℋ ) → B(ℋ ) be a n-normalized bounded positive linear map. In addition, by using the concept of quadruple D-synchronous functions which is generalizes the concept of a pair of synchronous functions, we establish an inequality similar to čebyšev inequality.

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.

Improved Young and Heinz operator inequalities for unitarily invariant norms

2020 ◽  
Vol 70 (2) ◽  
pp. 453-466
Author(s):  
A. Beiranvand ◽  
Amir Ghasem Ghazanfari
Keyword(s):  
Hilbert Space ◽  
Type Inequality ◽  
Young Inequality ◽  
Unitarily Invariant Norm ◽  
Operator Inequalities ◽  
Schmidt Norm ◽  
Unitarily Invariant Norms ◽  
Invariant Norm ◽  
Kantorovich Constant ◽  
New Inequalities

Abstract In this paper, we present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert-Schmidt norm of matrices. We also give some refinements of the following Heron type inequality for unitarily invariant norm |||⋅||| and A, B, X ∈ Mn(ℂ): $$\begin{array}{} \begin{split} \displaystyle \Big|\Big|\Big|\frac{A^\nu XB^{1-\nu}+A^{1-\nu}XB^\nu}{2}\Big|\Big|\Big| \leq &(4r_0-1)|||A^{\frac{1}{2}}XB^{\frac{1}{2}}||| \\ &+2(1-2r_0)\Big|\Big|\Big|(1-\alpha)A^{\frac{1}{2}}XB^{\frac{1}{2}} +\alpha\Big(\frac{AX+XB}{2}\Big)\Big|\Big|\Big|, \end{split} \end{array}$$ where $\begin{array}{} \displaystyle \frac{1}{4}\leq \nu \leq \frac{3}{4}, \alpha \in [\frac{1}{2},\infty ) \end{array}$ and r0 = min{ν, 1 – ν}.

Some remarks on a paper by W. N. Everitt

1977 ◽  
Vol 78 (1-2) ◽  
pp. 71-79 ◽  
Author(s):  
K. Daho ◽  
H. Langer
Keyword(s):  
Hilbert Space ◽  
Weight Function ◽  
Selfadjoint Operator ◽  
Krein Space ◽  
Indefinite Weight ◽  
Indefinite Weight Function ◽  
Kreĭn Space ◽  
Pontrjagin Space

Everitt has shown [1[, that for α ∊ [0, π/2] the undernoted problem (1.1–2) with an indefinite weight function r can be represented by a selfadjoint operator in a suitable Hilbert space. This result is extended to arbitrary α ∊ [0, π), replacing the Hilbert space in some cases by a Pontrjagin space with index one. The problem is also treated in the Krein space generated by the weight function r.

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