scholarly journals A numerical radius version of the arithmetic-geometric mean of operators

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2139-2145
Author(s):  
Alemeh Sheikhhosseini
Keyword(s):  
Geometric Mean ◽  
Positive Definite ◽  
Operator Norm ◽  
Numerical Radius ◽  
Positive Definite Operators ◽  
Inequalities For Operators

In this paper, we obtain some numerical radius inequalities for operators, in particular for positive definite operators A; B a numerical radius and some operator norm versions for arithmeticgeometric mean inequality are obtained, respectively as ?2(A#B)? ? (A2+B2/2)- 1/2inf ||x||=1 ?(x), where ?(x) = ?(A - B)x,x?2, and ||A||||B|| ? 1/2 (||A2||+||B2||)-1/2 inf ||x||=||y||=1 ?(x,y), where, ?(x,y) = (?Ay,y? - ?Bx,x?)2.

scholarly journals Some trace inequalities for operators

1995 ◽  
Vol 58 (2) ◽  
pp. 281-286 ◽  
Author(s):  
Xinmin Yang
Keyword(s):  
Positive Definite ◽  
Positive Definite Operators ◽  
Open Question ◽  
Trace Inequalities ◽  
Inequalities For Operators

AbstractIn this paper, we obtain some trace inequalities for arbitrary finite positive definite operators. Finally an open question is presented.

A note on the matrix arithmetic-geometric mean inequality

2018 ◽  
Vol 34 ◽  
pp. 283-287
Author(s):  
Teng Zhang
Keyword(s):  
Positive Integer ◽  
Geometric Mean ◽  
Positive Definite ◽  
Operator Norm ◽  
Positive Definite Matrices ◽  
The Matrix ◽  
Special Case

This note proves the following inequality: If $n=3k$ for some positive integer $k$, then for any $n$ positive definite matrices $\bA_1,\bA_2,\dots,\bA_n$, the following inequality holds: \begin{equation*}\label{eq:main} \frac{1}{n^3} \, \Big\|\sum_{j_1,j_2,j_3=1}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\| \,\geq\, \frac{(n-3)!}{n!} \, \Big\|\sum_{\substack{j_1,j_2,j_3=1,\\\text{$j_1$, $j_2$, $j_3$ all distinct}}}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\|, \end{equation*} where $\|\cdot\|$ represents the operator norm. This inequality is a special case of a recent conjecture proposed by Recht and R\'{e} (2012).

scholarly journals Some upper bounds for the Berezin number of Hilbert space operators

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4353-4360
Author(s):  
Ali Taghavi ◽  
Tahere Roushan ◽  
Vahid Darvish
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Geometric Mean ◽  
Upper Bounds ◽  
Positive Definite ◽  
Berezin Symbol ◽  
Hilbert Space Operators ◽  
Positive Definite Operators ◽  
Berezin Number ◽  
Definition Of

In this paper, we obtain some Berezin number inequalities based on the definition of Berezin symbol. Among other inequalities, we show that if A,B be positive definite operators in B(H), and A#B is the geometric mean of them, then ber2(A#B) ? ber (A2+B2/2)- 1/2 inf ????(?k); where ?(?k?) = ?(A-B)?k?,?k??2, and ?k? is the normalized reproducing kernel of the space H for ? belong to some set.

scholarly journals Maps on positive definite operators preserving the quantum $$\chi _\alpha ^2$$ χ α 2 -divergence

2017 ◽  
Vol 107 (12) ◽  
pp. 2267-2290 ◽  
Author(s):  
Hong-Yi Chen ◽  
György Pál Gehér ◽  
Chih-Neng Liu ◽  
Lajos Molnár ◽  
Dániel Virosztek ◽  
...  
Keyword(s):  
Positive Definite ◽  
Positive Definite Operators

Refinements of the heinz inequalities for operators and matrices

2018 ◽  
Vol 68 (6) ◽  
pp. 1431-1438
Author(s):  
Mahdi Mohammadi Gohari ◽  
Maryam Amyari
Keyword(s):  
Geometric Mean ◽  
Unitarily Invariant Norms ◽  
Heinz Mean ◽  
Inequalities For Operators ◽  
Class Of Functions

Abstract Suppose that A, B ∈ 𝔹(𝓗) are positive invertible operators. In this paper, we show that $$\begin{array}{} \displaystyle A \# B \leq \frac{1}{1-2\mu}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}\\ \displaystyle\qquad~\,\leq\frac{1}{2}\bigg[ A \# B +H_\mu (A,B)\bigg]\\ \displaystyle\qquad~\,\leq\frac{1}{2}\bigg[ \frac{1}{1-2\mu}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}+H_\mu (A,B)\bigg]\\ \displaystyle\qquad~\,\leq \dots \leq \frac{1}{2^n}A \# B + \frac{2^n-1}{2^n}H_\mu (A,B)\\ \displaystyle\qquad~\,\leq \frac{1}{2^n(1-2\mu)}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}+\frac{2^n-1}{2^n}H_\mu (A,B)\\ \displaystyle\qquad~\,\leq \frac{1}{2^{n+1}} A \# B +\frac{2^{n+1}-1}{2^{n+1}}H_\mu (A,B)\\ \displaystyle\qquad~\,\leq \dots \leq H_\mu (A,B) \end{array}$$ for each $\begin{array}{} \displaystyle \mu \in [0,1]\smallsetminus\{\frac{1}{2}\}, \end{array}$ where Hμ (A, B) and A#B are the Heinz mean and the geometric mean for operators A, B, respectively, and $\begin{array}{} \displaystyle F_{\mu}\in C({\rm sp}(A^\frac{-1}{2}BA^\frac{-1}{2})) \end{array}$ is a certain parameterized class of functions. As an application, we present several inequalities for unitarily invariant norms.

scholarly journals Some inequalities for the norm and the numerical radius of linear operators in Hilbert spaces

2008 ◽  
Vol 39 (1) ◽  
pp. 1-7 ◽  
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Operator Norm ◽  
Inner Product ◽  
Product Spaces ◽  
Numerical Radius ◽  
Inner Product Spaces

In this paper various inequalities between the operator norm and its numerical radius are provided. For this purpose, we employ some classical inequalities for vectors in inner product spaces due to Buzano, Goldstein-Ryff-Clarke, Dragomir-S ´andor and the author.

scholarly journals Geometric mean of partial positive definite matrices with missing entries

2019 ◽  
Vol 68 (12) ◽  
pp. 2408-2433
Author(s):  
Hayoung Choi ◽  
Sejong Kim ◽  
Yuanming Shi
Keyword(s):  
Geometric Mean ◽  
Positive Definite ◽  
Positive Definite Matrices

Sharp bounded for numerical radius and operator norm inequalities

2013 ◽  
Vol 7 ◽  
pp. 741-745
Author(s):  
H. Khosravi ◽  
M. Khanehgir ◽  
E. Faryad ◽  
P. Jafari
Keyword(s):  
Operator Norm ◽  
Numerical Radius ◽  
Norm Inequalities
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