Operator inequalities and characterizations

Advances in Operator Theory, 2023

In this survey, we shall present the characterizations of some distinguished classes of bounded linear operators acting on a complex separable Hilbert space in terms of operator inequalities related to the arithmetic–geometric mean inequality.

arXiv: Functional Analysis, 2020

In this survey, we shall present characterizations of some distinguished classes of bounded linear operators acting on a complex Hilbert space in terms of operator inequalities related to the arithmetic-geometric mean inequality.

Banach Journal of Mathematical Analysis, 2012

In the present paper, taking some advantages offered by the context of finite dimensional Hilbert spaces, we shall give a complete characterizations of certain distinguished classes of operators (self-adjoint, unitary reflection, normal) in terms of operator inequalities. These results extend previous characterizations obtained by the second author.

Publicationes Mathematicae Debrecen, 2012

The main objective of this paper is an improvement of the original weighted operator arithmetic-geometric mean inequality in Hilbert space. We define the difference operator between the arithmetic and geometric means, and investigate its properties. Due to the derived properties, we obtain a refinement and a converse of the observed operator mean inequality. As an application, we establish one significant operator mean, which interpolates the arithmetic and geometric means, that is, the Heinz operator mean. We also obtain an improvement of this interpolation.

Banach Journal of Mathematical Analysis, 2013

In this paper we derive some improvements of means inequalities for Hilbert space operators. More precisely, we obtain refinements and reverses of the arithmetic-geometric operator mean inequality. As an application, we also deduce an improved variant for the refined arithmetic-Heinz mean inequality. We also present some eigenvalue inequalities for differences of certain operator means.

Publications of the Research Institute for Mathematical Sciences, 1988

Several inequalities for Hilbert space operators are extended. These include results of Furuta, Halmos, and Kato on the mixed Schwarz inequality, the generalized Reid inequality as proved by Halmos and a classical inequality in the theory of compact non-self-adjoint operators which is essentially due to Weyl. Some related inequalities are also discussed.

Linear Algebra and its Applications, 1994

Porta, and Recht recently proved that (ISTS-' + S-'TSIj > 21jTI(. A generalization of this inequality to larger classes of operators and norms is obtained as an immediate consequence of the operator form of the arithmetic-geometric-mean inequality. Some related inequalities are also discussed. 1.

Banach Journal of Mathematical Analysis, 2015

In this paper some inequalities involving quasi-arithmetic means for a continuous field of self-adjoint operators, a field of positive linear mappings and continuous strictly monotone functions are refined. These refined converses are presented by using the Mond-Pečarić method improvement. Obtained results are applied to refine selected inequalities with power functions.

Linear and Multilinear Algebra, 2011

Let BðH Þ, IðH Þ and UðH Þ be the C Ã-algebra of all bounded linear operators acting on a complex Hilbert space H, the set of all invertible elements in BðH Þ and the class of all unitary operators in BðH Þ, respectively. In this note, we shall show that if S 2 IðH Þ, then the injective norm of S S À1 þ S À1 S in the tensor product space BðH Þ BðH Þ attains its minimal value 2 if and only if S is normal and satisfies the condition j þ j 2 for every , in the spectrum (S) of S. Finally, it is shown that if S 2 IðH Þ, then the inequality kSXS À1 þ S À1 XSk 2kXk holds for all X in BðH Þ if and only if S 2 R Ã UðH Þ.

Canadian Mathematical Bulletin, 1999

Let 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, ∞), then for 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.