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

scholarly journals More on -Normal Operators in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Rasoul Eskandari ◽  
Farzollah Mirzapour ◽  
Ali Morassaei
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Operator Norm ◽  
Numerical Radius ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Normal Operators

We study some properties of -normal operators and we present various inequalities between the operator norm and the numerical radius of -normal operators on Banach algebraℬ() of all bounded linear operators , where is Hilbert space.

scholarly journals Numerical Radius Inequalities for Several Operators

2014 ◽  
Vol 114 (1) ◽  
pp. 110 ◽  
Author(s):  
Omar Hirzallah ◽  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Operator Norm ◽  
Numerical Radius ◽  
Bounded Linear Operators ◽  
Bounded Linear

Let $A$, $B$, $X$, and $A_{1},\dots,A_{2n}$ be bounded linear operators on a complex Hilbert space. It is shown that \[ w\Bigl(\sum_{k=1}^{2n-1}A_{k+1}^{\ast}XA_{k}+A_{1}^{\ast}XA_{2n}\Bigr) \leq 2\Bigl( \sum_{k=1}^{n}\Vert A_{2k-1}\Vert^{2}\Bigr)^{1/2}\Bigl(\sum_{k=1}^{n}\left\Vert A_{2k}\right\Vert^{2}\Bigr)^{1/2}w(X) \] and \[ w(AB\pm BA)\leq 2\sqrt{2}\,\Vert B\Vert \sqrt{w^{2}(A)-\frac{\vert \Vert {\operatorname{Re} A}\Vert^{2}-\Vert {\operatorname{Im} A}\Vert^{2}\vert}{2}}, \] where $w(\cdot)$ and $\left\Vert \cdot \right\Vert$ are the numerical radius and the usual operator norm, respectively. These inequalities generalize and refine some earlier results of Fong and Holbrook. Some applications of our results are given.

scholarly journals States which have a trace-like property relative to a C*-subalgebra of B(H)

1976 ◽  
Vol 17 (2) ◽  
pp. 158-160
Author(s):  
Guyan Robertson
Keyword(s):  
Hilbert Space ◽  
Spectral Radius ◽  
Operator Theory ◽  
Linear Operators ◽  
Operator Norm ◽  
Identity Operator ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

In what follows, B(H) will denote the C*-algebra of all bounded linear operators on a Hilbert space H. Suppose we are given a C*-subalgebra A of B(H), which we shall suppose contains the identity operator 1. We are concerned with the existence of states f of B(H) which satisfy the following trace-like relation relative to A:Our first result shows the existence of states f satisfying (*), when A is the C*-algebra C*(x) generated by a normaloid operator × and the identity. This allows us to give simple proofs of some well-known results in operator theory. Recall that an operator × is normaloid if its operator norm equals its spectral radius.

scholarly journals Norm of a linear combination of two operators on a Hilbert space

2002 ◽  
Vol 65 (1) ◽  
pp. 9-22
Author(s):  
Takahiko Nakazi ◽  
Takanori Yamamoto
Keyword(s):  
Hilbert Space ◽  
Linear Combination ◽  
Linear Operators ◽  
Operator Norm ◽  
Complex Numbers ◽  
Bounded Linear Operators ◽  
Bounded Linear

Let α, β γ, δ be complex numbers such that γδ ≠ 0. If A and B are bounded linear operators on the Hilbert space H such that γA + δB is right invertible then we study the operator norm of (αA + βB)(γA + δB)−1 using the angle φ between two subspaces ran A and ran B or the angle ψ = ψ(A, B) between two operators A and B where

scholarly journals Joint numerical ranges: recent advances and applications minicourse by V. Müller and Yu. Tomilov

2020 ◽  
Vol 7 (1) ◽  
pp. 133-154
Author(s):  
V. Müller ◽  
Yu. Tomilov
Keyword(s):  
Hilbert Space ◽  
Operator Theory ◽  
Unit Vector ◽  
Linear Operators ◽  
Numerical Radius ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
Single Operator ◽  
Orbits Of Operators

AbstractWe present a survey of some recent results concerning joint numerical ranges of n-tuples of Hilbert space operators, accompanied with several new observations and remarks. Thereafter, numerical ranges techniques will be applied to various problems of operator theory. In particular, we discuss problems concerning orbits of operators, diagonals of operators and their tuples, and pinching problems. Lastly, motivated by known results on the numerical radius of a single operator, we examine whether, given bounded linear operators T1, . . ., Tn on a Hilbert space H, there exists a unit vector x ∈ H such that |〈Tjx, x〉| is “large” for all j = 1, . . . , n.

scholarly journals Exact and Approximate Operator Parallelism

2015 ◽  
Vol 58 (1) ◽  
pp. 207-224 ◽  
Author(s):  
Ali Zamani ◽  
Mohammad Sal Moslehian
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Operator Norm ◽  
Normed Spaces ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
C Algebra ◽  
Hilbert Space Operators ◽  
The Relationship

AbstractExtending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra B(ℋ) of bounded linear operators acting on a Hilbert space H . Among other things, we investigate the relationship between the approximate parallelism and norm of inner derivations on B(ℋ). We also characterize the parallel elements of a C*-algebra by using states. Finally we utilize the linking algebra to give some equivalent assertions regarding parallel elements in a Hilbert C*-module.

scholarly journals Some vector inequalities for two operators in Hilbert spaces with applications

2016 ◽  
Vol 8 (1) ◽  
pp. 75-92
Author(s):  
Sever S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Numerical Radius ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Nonnegative Operator

AbstractIn this paper we establish some vector inequalities for two operators related to Schwarz and Buzano results. We show amongst others that in a Hilbert space H we have the inequality $${1 \over 2}\left[ {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{x}},{\rm{x}}} \right\rangle ^{1/2} \left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{y}},{\rm{y}}} \right\rangle ^{1/2} + \left| {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over {\rm{2}}}} {\rm{x}},{\rm{y}}\right\rangle } \right|} \right] \ge \left| {\left\langle {{\mathop{\rm Re}\nolimits} ({\rm{B}}*{\rm{A}})\,{\rm{x}},{\rm{y}}} \right\rangle } \right|$$ for A, B two bounded linear operators on H such that Re (B*A) is a nonnegative operator and any vectors x, y ∈ H.Applications for norm and numerical radius inequalities are given as well.

scholarly journals Hypo-q-Norms on a Cartesian Product of Algebras of Operators on Banach Spaces

2017 ◽  
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Banach Spaces ◽  
Cartesian Product ◽  
Schwarz Inequality ◽  
Linear Operators ◽  
Operator Norm ◽  
Numerical Radius ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Inner Products ◽  
Reverse Inequalities

In this paper we introduce the hypo-q-operator norm and hypo-q-numerical radius on a Cartesian product of algebras of bounded linear operators on Banach spaces. A representation of these norms in terms of semi-inner products, the equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the scalar reverses of Cauchy-Buniakowski-Schwarz inequality are also given.

scholarly journals Hypo-q-Norms on a Cartesian Product of Algebras of Operators on Banach Spaces

2020 ◽  
Vol 34 (2) ◽  
pp. 169-192
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Banach Spaces ◽  
Cartesian Product ◽  
Schwarz Inequality ◽  
Linear Operators ◽  
Operator Norm ◽  
Numerical Radius ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Inner Products ◽  
Reverse Inequalities

AbstractIn this paper we consider the hypo-q-operator norm and hypo-q-numerical radius on a Cartesian product of algebras of bounded linear operators on Banach spaces. A representation of these norms in terms of semi-inner products, the equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the scalar reverses of Cauchy-Buniakowski-Schwarz inequality are also given.

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