scholarly journals A norm inequality for pairs of commuting positive semidefinite matrices

2015 ◽  
Vol 30 ◽  
Author(s):  
Koenraad Audenaert
Keyword(s):  
Positive Semidefinite ◽  
Norm Inequality ◽  
Unitarily Invariant Norm ◽  
Trace Norm ◽  
Simple Application ◽  
Positive Semidefinite Matrices ◽  
Schmidt Norm ◽  
Invariant Norm

For $k=1,\ldots,K$, let $A_k$ and $B_k$ be positive semidefinite matrices such that, for each $k$, $A_k$ commutes with $B_k$. We show that, for any unitarily invariant norm, \[ |||\sum_{k=1}^K A_kB_k||| \le ||| (\sum_{k=1}^K A_k)\;(\sum_{k=1}^K B_k)|||. \] The $K=2$ case was recently conjectured by Hayajneh and Kittaneh and proven by them for the trace norm and the Hilbert-Schmidt norm. A simple application of this norm inequality answers a question by Bourin in the affirmative.

Norm inequalities for positive semidefinite matrices and a question of Bourin

2017 ◽  
Vol 28 (14) ◽  
pp. 1750102 ◽  
Author(s):  
Mostafa Hayajneh ◽  
Saja Hayajneh ◽  
Fuad Kittaneh
Keyword(s):  
Positive Semidefinite ◽  
Affirmative Answer ◽  
Unitarily Invariant Norm ◽  
Trace Norm ◽  
Positive Semidefinite Matrices ◽  
Norm Inequalities ◽  
Invariant Norm

Let [Formula: see text] such that [Formula: see text] and [Formula: see text] are positive semidefinite. It is shown that [Formula: see text] for [Formula: see text] and for every unitarily invariant norm. This gives an affirmative answer to one of the questions posed by Bourin regarding subadditivity inequalities in the case of the trace norm. New norm inequalities related to Bourin's question are also presented.

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}.

scholarly journals A new young type inequality involving Heinz mean

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3639-3654
Author(s):  
Changsen Yang ◽  
Yu Li
Keyword(s):  
Type Inequality ◽  
Unitarily Invariant Norm ◽  
Operator Inequalities ◽  
Schmidt Norm ◽  
Invariant Norm ◽  
Kantorovich Constant ◽  
Heinz Mean

In this paper, we gave a new Young type inequality and the relevant Heinz mean inequality. Furthermore, we also improved some inequalities with Kantorovich constant or Specht?s ratio. Meanwhile, on the base of our scalars results, we obtain some new corresponding operator inequalities and matrix versions including Hilbert-Schmidt norm, unitarily invariant norm and related trace versions, which can be regarded as the application of our scalar results.

scholarly journals A trace bound for integer-diagonal positive semidefinite matrices

2020 ◽  
Vol 8 (1) ◽  
pp. 14-16
Author(s):  
Lon Mitchell
Keyword(s):  
Positive Semidefinite ◽  
Positive Semidefinite Matrix ◽  
Positive Semidefinite Matrices ◽  
Semidefinite Matrix

AbstractWe prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.

Norm inequalities for positive semi-definite matrices and a question of Bourin II

2021 ◽  
pp. 2150043
Author(s):  
Mostafa Hayajneh ◽  
Saja Hayajneh ◽  
Fuad Kittaneh
Keyword(s):  
Affirmative Answer ◽  
Unitarily Invariant Norm ◽  
Norm Inequalities ◽  
Invariant Norm ◽  
Special Case

Let [Formula: see text] and [Formula: see text] be [Formula: see text] positive semi-definite matrices. It is shown that [Formula: see text] for every unitarily invariant norm. This gives an affirmative answer to a question of Bourin in a special case. It is also shown that [Formula: see text] for [Formula: see text] and for every unitarily invariant norm.

scholarly journals Some Hermite-Hadamard type integral inequalities for operator AG-preinvex functions

2016 ◽  
Vol 8 (2) ◽  
pp. 312-323
Author(s):  
Ali Taghavi ◽  
Haji Mohammad Nazari ◽  
Vahid Darvish
Keyword(s):  
Integral Inequalities ◽  
Unitarily Invariant Norm ◽  
Norm Inequalities ◽  
Invariant Norm ◽  
Type Integral ◽  
Preinvex Functions ◽  
Inequalities For Operators

Abstract In this paper, we introduce the concept of operator AG-preinvex functions and prove some Hermite-Hadamard type inequalities for these functions. As application, we obtain some unitarily invariant norm inequalities for operators.

scholarly journals Fischer Type Log-Majorization of Singular Values on Partitioned Positive Semidefinite Matrices

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Benju Wang ◽  
Yun Zhang
Keyword(s):  
Singular Values ◽  
Positive Semidefinite ◽  
Positive Semidefinite Matrices ◽  
Fischer’S Inequality ◽  
New Inequalities

In this paper, we establish a Fischer type log-majorization of singular values on partitioned positive semidefinite matrices, which generalizes the classical Fischer's inequality. Meanwhile, some related and new inequalities are also obtained.

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