scholarly journals Some Improvements of the Cauchy-Schwarz Inequality Using the Tapia Semi-Inner-Product

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2112
Author(s):  
Nicuşor Minculete ◽  
Hamid Reza Moradi
Keyword(s):  
Hilbert Space ◽  
Normed Space ◽  
Triangle Inequality ◽  
Schwarz Inequality ◽  
Inner Product ◽  
Numerical Radius ◽  
Real Numbers ◽  
Norm Inequalities ◽  
Hilbert Space Operators ◽  
New Inequalities

The aim of this article is to establish several estimates of the triangle inequality in a normed space over the field of real numbers. We obtain some improvements of the Cauchy–Schwarz inequality, which is improved by using the Tapia semi-inner-product. Finally, we obtain some new inequalities for the numerical radius and norm inequalities for Hilbert space operators.

scholarly journals About the Cauchy–Bunyakovsky–Schwarz Inequality for Hilbert Space Operators

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 305
Author(s):  
Nicuşor Minculete
Keyword(s):  
Hilbert Space ◽  
Operator Theory ◽  
Schwarz Inequality ◽  
Linear Operators ◽  
Real Numbers ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
Bohr’S Inequality ◽  
Symmetric Shape

The symmetric shape of some inequalities between two sequences of real numbers generates inequalities of the same shape in operator theory. In this paper, we study a new refinement of the Cauchy–Bunyakovsky–Schwarz inequality for Euclidean spaces and several inequalities for two bounded linear operators on a Hilbert space, where we mention Bohr’s inequality and Bergström’s inequality for operators. We present an inequality of the Cauchy–Bunyakovsky–Schwarz type for bounded linear operators, by the technique of the monotony of a sequence. We also prove a refinement of the Aczél inequality for bounded linear operators on a Hilbert space. Finally, we present several applications of some identities for Hermitian operators.

scholarly journals Operator equalities and Characterizations of Orthogonality in Pre-Hilbert C*-Modules

2021 ◽  
pp. 1-21
Author(s):  
Rasoul Eskandari ◽  
M. S. Moslehian ◽  
Dan Popovici
Keyword(s):  
Hilbert Space ◽  
Triangle Inequality ◽  
Sufficient Conditions ◽  
Negative Real Part ◽  
Necessary And Sufficient Conditions ◽  
Inner Product ◽  
Equality Case ◽  
Hilbert Space Operators ◽  
Necessary And Sufficient ◽  
The Relationship

Abstract In the first part of the paper, we use states on $C^{*}$ -algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^{*}$ -module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^{*}$ -module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^{*}$ -module under the assumption that their inner product has a negative real part. We introduce the concept of Pythagoras orthogonality and discuss its properties. We describe this notion for Hilbert space operators in terms of the parallelogram law and some limit conditions. We present several examples in order to illustrate the relationship between the Birkhoff–James, Roberts, and Pythagoras orthogonalities, and the usual orthogonality in the framework of Hilbert $C^{*}$ -modules.

IMPROVED INEQUALITIES FOR THE NUMERICAL RADIUS: WHEN INVERSE COMMUTES WITH THE NORM

2018 ◽  
Vol 97 (2) ◽  
pp. 293-296 ◽  
Author(s):  
BRYAN E. CAIN
Keyword(s):  
Hilbert Space ◽  
Numerical Radius ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
New Inequalities

New inequalities relating the norm $n(X)$ and the numerical radius $w(X)$ of invertible bounded linear Hilbert space operators were announced by Hosseini and Omidvar [‘Some inequalities for the numerical radius for Hilbert space operators’, Bull. Aust. Math. Soc.94 (2016), 489–496]. For example, they asserted that $w(AB)\leq$$2w(A)w(B)$ for invertible bounded linear Hilbert space operators $A$ and $B$. We identify implicit hypotheses used in their discovery. The inequalities and their proofs can be made good by adding the extra hypotheses which take the form $n(X^{-1})=n(X)^{-1}$. We give counterexamples in the absence of such additional hypotheses. Finally, we show that these hypotheses yield even stronger conclusions, for example, $w(AB)=w(A)w(B)$.

scholarly journals FURTHER INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS

2021 ◽  
Vol 12 (4) ◽  
pp. 25-32
Author(s):  
HASSAN RANJBAR ◽  
ASADOLLAH NIKNAM
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Operator Norm ◽  
Numerical Radius ◽  
Bounded Linear Operators ◽  
Norm Inequalities ◽  
Bounded Linear ◽  
Hermitian Forms ◽  
Sum Of Products

By use of some non-negative Hermitian forms defined for n-tuple of bounded linear operators on the Hilbert space (H, h·, ·i) we establish new numerical radius and operator norm inequalities for sum of products of operators

Sign in / Sign up

Export Citation Format

Share Document