scholarly journals Some Inequalities of the Grüss Type for the Numerical Radius of Bounded Linear Operators in Hilbert Spaces

2008 ◽  
Vol 2008 ◽  
pp. 1-9 ◽  
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Numerical Radius ◽  
Bounded Linear Operators ◽  
Bounded Linear

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 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.

Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

scholarly journals WEYL TYPE THEOREMS FOR FUNCTIONS OF OPERATORS

2012 ◽  
Vol 54 (3) ◽  
pp. 493-505 ◽  
Author(s):  
SEN ZHU ◽  
CHUN GUANG LI ◽  
TING TING ZHOU
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Compact Perturbations ◽  
Weyl Type Theorems ◽  
Necessary And Sufficient

AbstractA-Weyl's theorem and property (ω), as two variations of Weyl's theorem, were introduced by Rakočević. In this paper, we study a-Weyl's theorem and property (ω) for functions of bounded linear operators. A necessary and sufficient condition is given for an operator T to satisfy that f(T) obeys a-Weyl's theorem (property (ω)) for all f ∈ Hol(σ(T)). Also we investigate the small-compact perturbations of operators satisfying a-Weyl's theorem (property (ω)) in the setting of separable Hilbert spaces.

Sign in / Sign up

Export Citation Format

Share Document