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 Improved Young and Heinz operator inequalities with Kantorovich constant

2021 ◽  
Vol 73 (1) ◽  
pp. 23-32
Author(s):  
A. Beiranvand ◽  
A. G. Ghazanfari
Keyword(s):  
Hilbert Space ◽  
Young Inequality ◽  
Operator Inequalities ◽  
Schmidt Norm ◽  
Kantorovich Constant ◽  
New Inequalities

UDC 517.9 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.  

Some refined Young type inequalities using different weights

2021 ◽  
Author(s):  
Leila Nasiri ◽  
Bahman Askari
Keyword(s):  
Type Inequality ◽  
Matrix Inequalities ◽  
Real Numbers ◽  
Positive Real ◽  
Unitarily Invariant Norms ◽  
New Inequalities

The Young type inequality asserts that if [Formula: see text] are two positive real numbers, then for each [Formula: see text], we have [Formula: see text] In this paper, we obtain some new inequalities using two different weights. For example, if [Formula: see text], then [Formula: see text] [Formula: see text] where [Formula: see text] In addition, we refine some matrix inequalities for Unitarily invariant norms by applying the deduced numerical inequalities.

Some Norm Inequalities for Operators

1999 ◽  
Vol 42 (1) ◽  
pp. 87-96 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Geometric Mean ◽  
Continuous Functions ◽  
Unitarily Invariant Norm ◽  
Norm Inequalities ◽  
Invariant Norm ◽  
Inequalities For Operators

AbstractLet Ai , Bi and Xi (i = 1, 2,…,n) be operators on a separable Hilbert space. It is shown that if f and g are nonnegative continuous functions on [0, ∞) which satisfy the relation f(t)g(t) = t for all t in [0, ∞), thenfor every r > 0 and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.

scholarly journals On the convexity of functions

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3773-3781
Author(s):  
Ata Abu-Asad ◽  
Omar Hirzallah
Keyword(s):  
Hilbert Space ◽  
Real Number ◽  
Positive Real Number ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Unitarily Invariant Norm ◽  
Positive Real ◽  
Bounded Linear ◽  
Invariant Norm ◽  
Increasing Function

Let A,B, and X be bounded linear operators on a separable Hilbert space such that A,B are positive, X ? ?I, for some positive real number ?, and ? ? [0,1]. Among other results, it is shown that if f(t) is an increasing function on [0,?) with f(0) = 0 such that f(?t) is convex, then ?|||f(?A + (1-?)B) + f(?|A-B|)|||?|||?f(A)X + (1-?)Xf (B)||| for every unitarily invariant norm, where ? = min (?,1-?). Applications of our results are given.

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.

scholarly journals Some inequalities for trace class operators via a Kato’s result

2017 ◽  
Vol 11 (01) ◽  
pp. 1850004
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Normal Operators ◽  
Trace Class Operators ◽  
Kato’S Inequality ◽  
New Inequalities

By the use of the celebrated Kato’s inequality, we obtain in this paper some new inequalities for trace class operators on a complex Hilbert space [Formula: see text] Natural applications for functions defined by power series of normal operators are given as well.

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.

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