Heinz Inequalities Research Articles

Refinements of the Heinz inequalities

This article aims to discuss Heinz inequalities involving unitarily invariant norms. We obtain refinements of the Heinz inequalities. In particular, our results refine some results given in Kittaneh.

Some inequalities for unitarily invariant norms

This paper aims to present some inequalities for unitarily invariant norms. In section 2, we give a refinement of the Cauchy-Schwarz inequality for matrices. In section 3, we obtain an improvement for the result of Bhatia and Kittaneh (Linear Algebra Appl. 308 (2000) 203-211). In section 4, we establish an improved Heinz inequality for the Hilbert-Schmidt norm. Finally, we present an inequality involving positive definite matrix and Hilbert-Schmidt norm. Then we use it to discuss the conjecture on the Hilbert-Schmidt norm of matrices proposed by Sloane and Harwit and the conjecture is proved for some special matrices.

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 for matrices. J. Math. Anal. Appl.361(2010)262-269]. Then we present a new inequality for unitarily invariant norms, which is equivalent to an inequality presented by Kittaneh and Manasrah in the case of the Hilbert-Schmidt norm.

Operator inequalities related to the Corach–Porta–Recht inequality

We prove some refinements of an inequality due to X. Zhan in an arbitrary complex Hilbert space by using some results on the Heinz inequality. We present several related inequalities as well as new variants of the Corach–Porta–Recht inequality. We also characterize the class of operators satisfying SXS-1+S-1XS+kX⩾(k+2)X under certain conditions.

Reverse Young and Heinz inequalities for matrices

We give reverses of the classical Young inequality for positive real numbers and we use these to establish reverse Young and Heinz inequalities for matrices.

MEAN THEORETIC OPERATOR FUNCTIONS FOR EXTENSIONS OF THE GRAND FURUTA INEQUALITY

The Furuta inequality is known as an exquisite extension of the Lowner- Heinz inequality. The grand Furuta inequality is its further extension including Ando- Hiai inequality on log-majorization. We introduce two mean theoretic operator functions defined for positive invertible operators X ≤ A, for a given positive invertible operator A and given t ∈ (0, 1) : Φ(X)(= Φ(u, p; X)) = Auu+1 u+p X p (≤ X) for u ≥ 0 ,p ≥ 1 and Ψ(X)(= Ψ(q, s; X)) = (A tsX q ) 1 (1−s)t+sq (≤ X) for q ≥ 1 ,s ≥ 1. We have the grand Furuta inequality by using their composition and also have its further extension presented recently by Furuta, by using a successive composition of them.

Double Operator Integrals and Submajorization

We present a user-friendly version of a double operator integration theory which still retains a capacity for many useful applications. Using recent results from the latter theory applied in noncommutative geometry, we derive applications to analogues of the classical Heinz inequality, a simplified proof of a famous inequality of Birman-Koplienko-Solomyak and also to the Connes-Moscovici inequality. Our methods are sufficiently strong to treat these inequalities in the setting of symmetric operator norms in general semifinite von Neumann algebras.

Improved Young and Heinz inequalities for matrices

We give refinements of the classical Young inequality for positive real numbers and we use these refinements to establish improved Young and Heinz inequalities for matrices.

Norm inequalities in operator ideals

In this paper we introduce a new technique for proving norm inequalities in operator ideals with a unitarily invariant norm. Among the well-known inequalities which can be proved with this technique are the Löwner–Heinz inequality, inequalities relating various operator means and the Corach–Porta–Recht inequality. We prove two general inequalities and from them we derive several inequalities by specialization, many of them new. We also show how some inequalities, known to be valid for matrices or bounded operators, can be extended with this technique to normed ideals in C ∗ -algebras, in particular to the noncommutative L p -spaces of a semi-finite von Neumann algebra.

On Bernstein–Heinz–Chern–Flanders inequalities

AbstractWe give an interpretation of the Chern–Heinz inequalities for graphs in order to extend them to transversally oriented codimension one C2-foliations of Riemannian manifolds. It contains Salavessa's work on mean curvature of graphs and fully generalizes results of Barbosa–Kenmotsu–Oshikiri [3] and Barbosa–Gomes–Silveira [2] about foliations of 3-dimensional Riemannian manifolds by constant mean curvature surfaces. This point of view of the Chern–Heinz inequalities can be applied to prove a Haymann–Makai–Osserman inequality (lower bounds of the fundamental tones of bounded open subsets Ω ⊂ ℝ2 in terms of its inradius) for embedded tubular neighbourhoods of simple curves of ℝn.

On High-power Operator Inequalities and Spectral Radii of Operators

For some different types of operators on a Hilbert space, we present new high-power operator inequalities, and their corresponding operator inequalities involving spectral radii of operators. In particular, we show that Halmos’ two operator inequalities, Reid’s inequality and many others hold easily. In applications we obtain a new generalized classical Löwner inequality; and a slightly generalized Löwner–Heinz inequality is given.

Characterizations of logA?logB and normaloid operators via Heinz inequality

In 1951, Heinz showed the following useful norm inequality:“If A, B≥0and X∈B(H), then ‖AXB‖ r ‖X‖1−r≥‖A r XB r ‖holds for r∈ [0, 1].” In this paper, we shall show the following two applications of this inequality:

Operator Monotone Functions which Are Defined Implicitly and Operator Inequalities

The function tα (0<α<1) is operator monotone on 0⩽t<∞. This is known as the Löwner–Heinz inequality. However, not too many examples of concrete operator monotone functions are known so far. We will systematically seek operator monotone functions which are defined implicitly. This investigation is new, and our method seems to be powerful. We will actually find a family of operator monotone functions which includes tα (0<α<1). Moreover, by constructing one-parameter families of operator monotone functions, we will get many operator inequalities; especially, we will extend the Furuta inequality and the exponential inequality of Ando.

Some Norm Inequalities for Operators

AbstractLet 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, ∞), thenfor every r &gt; 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.

Operator versions of inequalities and equalities on a Hilbert space

A study of sharpening Heinz inequality leads to definitions of weak operator versions and operator versions of inequalities. It is shown that fundamental inequalities of weak operator versions and operator versions in a Hilbert space are equivalent to one another. In particular, some characterizations of the Cauchy-Schwarz inequality are given.

Heinz’s inequality and Bernstein’s inequality

The purpose of the present account is to sharpen Heinz's inequality, and to investigate the equality and the bound of the inequality. As a consequence of this we present a Bernstein type inequality for nonselfadjoint operators. The Heinz inequality can be naturally extended to a more general case, and from which we obtain in particular Bessel's equality and inequality. Finally, Bernstein's inequality is extended to n eigenvectors, and shows that the bound of the inequality is preserved. The well-known Heinz inequality is as follows: The relation H8 I(Tx,y) 12 < (IT 12ax x)(IT* 12(1-,)y, y) holds for any bounded linear operator T on a complex Hilbert space H, x, y E H, and any real number oa with 0 < a < 1, where IT] is the positive square root of the operator T*T. It is possible to sharpen the inequality (*) if T*y is orthogonal to a vector z with Tz $4 0. The new inequality is naturally extended to a more general case when T*y is orthogonal to a set of vectors in which the bound of the inequality is retained. In particular we obtain Bessel's equality. By a similar method we present a Bernstein type inequality for nonselfadjoint operators. Finally, Bernstein's inequality is generalized to n eigenvectors for a selfadjoint operator and shows that the bound is preserved. Theorem 1. Let T be a bounded linear operator on a complex Hilbert space H and 0 5$ y E H. If T*y is orthogonal to a vector z E H with Tz $ 0, then I(TX I y)12 + 2( T*2(I -a)y, y)(IT] 2a X, Z) 1j2 < (IT I2X, x) (IT* 12(1-,a) If XY)I + ~~(IT12cZ, Z)-( Ixx y for every x E H and a E [0,1]. The equality holds if and only if the two vectors T*y and IT 12caX_(IT )I(ITI12 a z) are proportional, equivalently, the two vectors Tx (jTjl2xz)Tz and IT*12(1)y are proportional for 0 < a < 1. (ITI 2 Z, z) ITI()y pootnafr Received by the editors September 7, 1995 and, in revised form, September 19, 1995 and February 5, 1996. 1991 Mathematics Subject Classification. Primary 47A30, 65F15; Secondary 65J10. ?)1997 Arnerican Mathematical Society 2319 This content downloaded from 207.46.13.113 on Thu, 06 Oct 2016 04:34:08 UTC All use subject to http://about.jstor.org/terms

Rota's theorem and Heinz inequalities

The Heinz inequality and the Heinz-Kato inequality have several equivalent formulations as considered in our preceding notes. Based on these, we discuss the position of Rota's theorem, which is shown to imply the Heinz inequalities.

A note on the arithmetic-geometric mean inequality for every unitarily invariant matrix norm

We integrate ten unitarily invariant matrix norm inequalities equivalent to the Heinz inequality.

Norm Inequalities Equivalent to Heinz Inequality

We investigate several norm inequalities equivalent to the Heinz inequality and discuss the equivalence relations among these norm inequalities. Here we shall show an elementary and simplified proof to the famous Heinz inequality.

Norm inequalities equivalent to Heinz inequality

We investigate several norm inequalities equivalent to the Heinz inequality and discuss the equivalence relations among these norm inequalities. Here we shall show an elementary and simplified proof to the famous Heinz inequality.