Furuta Inequality Research Articles

J-selfadjoint matrix means and their indefinite inequalities

AbstractLet J be a non trivial involutive Hermitian matrix. Consider $${\mathbb {C}}^n$$ C n equipped with the indefinite inner product induced by J, $$[x,y]=y^*J x$$ [ x , y ] = y ∗ J x for all $$x,y\in {{\mathbb {C}}}^n,$$ x , y ∈ C n , which endows the matrix algebra $${\mathbb {C}}^{n\times n}$$ C n × n with a partial order relation $$\le ^J$$ ≤ J between J-selfadjoint matrices. Inde-finite inequalities are given in this setup, involving the J-selfadjoint $$\alpha $$ α -weighted geometric matrix mean. In particular, an indefinite version of Ando–Hiai inequality is proved to be equivalent to Furuta inequality of indefinite type.

Refinements of Some Numerical Radius Inequalities for Hilbert Space Operators

Some power inequalities for the numerical radius based on the recent Dragomir extension of Furuta's inequality are established. Some particular cases are also provided. Moreover, we get an improvement of the H\"older-McCarthy operator inequality in the case when $r\geq 1$ and refine generalized inequalities involving powers of the numerical radius for sums and products of Hilbert space operators.

SPECTRAL ORDER OF OPERATORS AND FURUTA INEQUALITY

By virtue of Furuta inequality, we show some characterizations of the spectral order of positive operators on a Hilbert space from the viewpoint of the Kan- torovich type inequality and the Riccati equation. Let A and B be positive operators on a Hilbert space. Then the spectral order AB holds if and only if there exist a unique positive contraction Tp,u such that Tp,uA p+u 2 Tp,u = A up 4 B p A up 4 for all p ≥ 0 and u ≥ 0. This form interpolates the chaotic order, δ-order and the usual order continuously. 1 Introduction. An operator means a bounded linear operator on a Hilbert space H. An operator A is said to be positive ( denoted by A ≥ 0) if (Ax, x) ≥ 0 for all x ∈ H.I n particular, we denote by A> 0i fA is positive and invertible. The usual order A ≥ B for selfadjoint operators A and B is defined by A − B ≥ 0. Let A and B be positive invertible operators. We introduce a function order A ≥f B for a real valued continuous function f on (0, ∞) defined by A ≥f B ⇐⇒ f(A) ≥ f(B).

Some log-majorizations and an extension of a determinantal inequality

An eigenvalue inequality involving a matrix connection and its dual is established, and some log-majorization type results are obtained. In particular, some eigenvalues inequalities considered by F. Hiai and M. Lin [9], an associated conjecture, and a singular values inequality by L. Zou [20] are revisited. A reformulation of the inequality det(A+U⁎B)≤det(A+B), for positive semidefinite matrices A,B, with U a unitary matrix that appears in the polar decomposition of BA, is also extended, using some known norm inequalities, associated to Furuta inequality and Araki–Cordes inequality.

Estimates of operator convex and operator monotone functions on bounded intervals

Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but also for operator convex functions on bounded intervals. More precisely, we prove that if $f$ is a nonlinear operator convex function on a bounded interval $(a,b)$ and $A, B$ are bounded linear operators acting on a Hilbert space with spectra in $(a,b)$ and $A-B$ is invertible, then $sf(A)+(1-s)f(B)>f(sA+(1-s)B)$. A short proof for a similar known result concerning a nonconstant operator monotone function on $[0,\infty)$ is presented. Another purpose is to find a lower bound for $f(A)-f(B)$, where $f$ is a nonconstant operator monotone function, by using a key lemma. We also give an estimation of the Furuta inequality, which is an excellent extension of the L\"owner--Heinz inequality.

Extension of Furuta inequality with nonnegative powers for multi-operator

Extension of Furuta inequality with nonnegative powers for multi-operator

CHARACTERIZATIONS OF OPERATOR ORDER FOR THREE POSITIVE DEFINITE OPERATORS VIA OPERATOR MEAN

Motivated by Lin and Cho's characterizations of A ≥ B ≥ C via extended grand Furuta inequality, we present two characterizations of A ≥ B ≥ C via operator mean.

Operator inequalities: From a general theorem to concrete inequalities

The aim of this paper is to give a method to extract concrete inequalities from a general theorem, which is established by making use of majorization relation between functions. By this method we can get a lot of inequalities; among others we extend Furuta inequality as follows: Let fi, gj be positive operator monotone functions on [0,∞) and put k(t)=tr0f1(t)r1⋯fm(t)rm, h(t)=tp0g1(t)p1⋯gn(t)pn, where p0≥1 and ri≥0, pj≥0. Then 0≤A≤C≤B implies, for 0<α≤1+r0p+r0 with p=p0+⋯+pn, (k(C)12h(A)k(C)12)α≤(k(C)12h(C)k(C)12)α≤(k(C)12h(B)k(C)12)α. Moreover, we show log⁡C1/2eAC1/2≤log⁡C1/2eCC1/2≤log⁡C1/2eBC1/2, provided C is invertible. We also refer to operator geometric mean.

An application of grand Furuta inequality to a type of operator equation

The existence of positive semidefinite solutions of the operator equation n X j=1 A n−j XA j−1 = Y is investigated by applying grand Furuta inequality. If there exists positive semidefinite solutions of the operator equation, one of the special types of Y is obtained, which extends the related result before. Finally, an example is given based on our result.

On the range of the parameters for the grand Furuta inequality to be valid

In order to investigate the precise range D of the parameters p, s, t, and r for which the grand Furuta inequality is valid. We use the method of reductio ad absurdum. We find the following results: The area , , , and is contained in the complement of the range D. The condition seems essential for the grand Furuta inequality. MSC:47A63, 15A45.

Some ways of constructing Furuta-type inequalities

Let σh be an operator mean associated with an operator monotone function h and let A,B be positive operators. We investigate the following Furuta-type inequality: For some fixed continuous function h,A≥B>0⇒A−rσφrXh−r≤B(r≥1), where Xh is the positive solution of h(X)=B−1. The map (h,r)↦φr plays a central role in constructing a Furuta-type inequality, similar to the role of the map (p,r)↦r+1p+r in the following part of the Furuta inequality: A≥B>0,p,r≥1⇒(Br2ApBr2)1+rp+r≥(Bp+r)1+rp+r. We obtain this map (h,r)↦φr from the following Ando–Hiai type inequality:A,B>0,AσhB≤1⇒ArσhBr≤1(r≥1).

Matrix inequalities including grand Furuta inequality via Karcher mean

In our previous paper, we have shown a generalization of Furuta inequality via Karcher mean (Riemannian mean) by using Yamazaki's results which are generalizations of Ando-Hiai inequality and related ones. In this paper, we shall show a generalization of grand Furuta in- equality as an extension of our previous result.

Some Properties of Furuta Type Inequalities and Applications

This work is to consider Furuta type inequalities and their applications. Firstly, some Furuta type inequalities underA≥B≥0are obtained via Loewner-Heinz inequality; as an application, a proof of Furuta inequality is given without using the invertibility of operators. Secondly, we show a unified satellite theorem of grand Furuta inequality which is an extension of the results by Fujii et al. At the end, a kind of Riccati type operator equation is discussed via Furuta type inequalities.

Some inequalities for measurable operators

In this paper, Loewner-Heinz inequality, Hansen inequality and Furuta inequality for measurable operators affiliated with a given von Neumann algebras will be given.

A certain functional inequality derived from an operator inequality

We will show a certain functional inequality involving fractional powers, making use of the grand Furuta inequality and Tanahashi's argument.

Generalizations of Furuta's inequality

We obtain amongst others the following result for any x, y ∈ H, where is a function defined by power series with real coefficients and convergent on the open disk D(0, R) ⊂ ℂ, R > 0, , T ∈ ℬ(H), α, β ≥ 0 with α + β ≥ 1 and ‖T ‖2α, ‖T‖2β < R. For constant functions this produces Furuta's inequality which in its turn is a generalization of Kato's inequality that is obtained for α ∈ [0, 1] and β = 1 − α.

On a relation between Schur, Hardy-Littlewood-Pólya and Karamata’s theorem and an inequality of some products of x p − 1 derived from the Furuta inequality

We show a functional inequality of some products of x p −1 as an application of an operator inequality. Furthermore, we will show it can be deduced from a classical theorem on majorization and convex functions.MSC:26D07, 26A09, 26A51, 39B62, 47A63.

Some Operator Inequalities on Chaotic Order and Monotonicity of Related Operator Function

We will discuss some operator inequalities on chaotic order about several operators, which are generalization of Furuta inequality and show monotonicity of related Furuta type operator function.

Another consequence of tanahashi’s argument on best possibility of the grand Furuta inequality

Abstract We show that Tanahashi’s argument on best possibility of the grand Furuta inequality has an additional consequence.

An application of matrix inequalities to certain functional inequalities involving fractional powers

We will show certain functional inequalities involving fractional powers, making use of the Furuta inequality and Tanahashi’s argument. MSC:26D07, 26A09, 39B62, 47A63.