Arithmetic–Geometric Mean and Related Inequalities for Operators

  title={Arithmetic–Geometric Mean and Related Inequalities for Operators},
  author={H. Kosaki},
  journal={Journal of Functional Analysis},
  • H. Kosaki
  • Published 1998
  • Mathematics
  • Journal of Functional Analysis
Abstract In recent years certain arithmetic–geometric mean and related inequalities for operators and unitarily invariant norms have been obtained by many authors based on majorization technique and so on. We first point out that they are direct consequences of integral expressions of relevant operators. Furthermore we obtain related new inequalities (Theorems 4, 5, and 6) based on our current approach. 
On improved arithmetic-geometric mean and Heinz inequalities for matrices
In this paper, we first generalize an inequality and improve another one for unitarily invariant norms, which are established by Kittaneh and Manasrah in [Improved Young and Heinz inequalities forExpand
Arithmetic–Geometric Mean and Related Submajorisation and Norm Inequalities for $$\tau $$-Measurable operators: Part I
The paper establishes arithmetic-geometric mean and related submajorisation and norm inequalities in the general setting of $$\tau $$ -measurable operators affiliated with a semi-finite von NeumannExpand
Singular value and arithmetic-geometric mean inequalities for operators
A singular value inequality for sums and products of Hilbert space operators is given. This inequality generalizes several recent singular value inequalities, and includes that if A, B, and X areExpand
Interpolating the arithmetic-geometric mean inequality and its operator version
Two families of means (called Heinz means and Heron means) that interpolate between the geometric and the arithmetic mean are considered. Comparison inequalities between them are established.Expand
Sharp Inequalities for Some Operator Means
  • D. Drissi
  • Mathematics, Computer Science
  • SIAM J. Matrix Anal. Appl.
  • 2006
In this paper sharp results on strong domination between the Heinz and logarithmic means are obtained. This leads to sharp operator inequalities extending results given by Bhatia-Davis andExpand
The Improved Arithmetic-Geometric Mean Inequalities for Matrix Norms
In this paper, we prove a scalar inequality such that this inequality is an improvement of the classical arithmetic-geometric mean inequality. We obtain its matrix version and investigateExpand
Further interpolation inequalities related to arithmetic-geometric mean, Cauchy-Schwarz and Hölder inequalities for unitarily invariant norms
An inequality for matrices that interpolates between the Cauchy-Schwarz and the arithmetic-geometric mean inequalities for unitarily invariant norms has been obtained by Audenaert. Alakhrass obtainedExpand
Notes on matrix arithmetic–geometric mean inequalities
Abstract For positive semi-definite n×n matrices, the inequality 4|||AB|||⩽|||(A+B) 2 ||| isshown to hold for every unitarily invariant norm. The connection of this with some other matrixExpand
More accurate operator means inequalities
Abstract Our main target in this paper is to present new sharp bounds for inequalities that result when weighted operator means are filtered through positive linear maps and operator monotoneExpand
Some inequalities involving unitarily invariant norms
This paper aims to present some inequalities for unitarily invariant norms. We first give inverses of Young and Heinz type inequalities for scalars. Then we use these inequalities to establish someExpand


Norm bounds for Hadamard products and an arithmetic - geometric mean inequality for unitarily invariant norms
Abstract An arithmetic-geometric mean inequality for unitarily invariant norms and matrices, 2∥A∗XB∥⩽∥AA∗X+XBB∗∥ , is an immediate consequence of a basic inequality for singular values of HadamardExpand
On Some Operator Inequalities
Abstract For Hilbert-space operators S , T with S invertible and self-adjoint, Corach, Porta, and Recht recently proved that ‖ STS -1 + S -1 TS ‖⩾2‖ T ‖. A generalization of this inequality to largerExpand
Log-majorizations and norm inequalities for exponential operators
Concise but self-contained reviews are given on theories of majorization and symmetrically normed ideals, including the proofs of the Lidskii–Wielandt and the Gelfand– Naimark theorems. Based onExpand
More matrix forms of the arithmetic-geometric mean inequality
For arbitrary $n \times n$ matrices A, B, X, and for every unitarily invariant norm, it is proved that $2|||A^ * XB|||\leqq |||AA^ * X + XBB^ * |||$.
Norm inequalities for fractional powers of positive operators
It is shown that ifA, B andX are operators on a Hilbert space such thatA andB are positive andX belongs to a norm ideal associated with some unitarily invariant norm |‖·|‖, then for 0 ≤r ≤ 1 we haveExpand
Introduction to Fourier Analysis on Euclidean Spaces.
The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the actionExpand
Abstract For any unitarily invariant norm on Hilbert-space operators it is shown that for all operators A , B , X and positive real numbers r we have ||| |A∗XB| r ||| 2 ⩽ ||| |AA∗X| r ||| ||| |XBB∗|Expand
Majorizations and inequalities in matrix theory
Abstract In matrix theory, majorization plays a significant role. For instance, majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and evenExpand
On the singular values of a product of operators
For compact Hilbert space operators A and B, the singular values of $A^ * B$ are shown to be dominated by those of $\frac{1}{2}(AA^* + BB^* )$.
An arithmetic-geometric-harmonic mean inequality involving Hadamard products
Abstract Given matrices of the same size, A = [ a ij ] and B = [ b ij ], we define their Hadamard product to be A ∘ B = [ a ij b ij ]. We show that if x i > 0 and q ⩾ p ⩾ 0, then the n × n matrices xExpand