Norm equalities and inequalities for three circulant operator matrices

In this paper, three new circulant operator matrices, scaled circulant operator matrices, diag-circulant operator matrices and retrocirculant operator matrices, are given respectively. Several norm equalities and inequalities for these operator matrices are proved. We show the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norm. Pinching type inequality is also given for weakly unitarily invariant norms. These results are closely related to the nice structure of these special operator matrices. Furthermore, some special cases and specific examples are also considered.     

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 Lp-spaces of a semi-finite von Neumann algebra.     

Norm inequalities for matrix geometric means of positive definite matrices

We obtain eigenvalue inequalities for matrix geometric means of positive definite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to a series of norm inequalities among geometric means. We give complements of the Ando–Hiai type inequality for the Karcher mean by means of the generalized Kantorovich constant. Finally, we consider the monotonicity of the eigenvalue function for the Karcher mean.     

Operator-norm inequalities and interpolated trace inequalities

In this article, we investigate some operator-norm inequalities related to some conjectures posed by Hayajneh and Kittaneh that are related to questions of Bourin regarding a special type of inequalities referred to as subadditivity inequalities. While some inequalities are meant to answer these conjectures, other inequalities present reverse-type inequalities for these conjectures. Then, we present some new trace inequalities related to Heinz means inequality and use these inequalities to prove some variants of the aforementioned conjectures.     

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.     

Numerical radius inequalities for operator matrices

We prove a numerical radius inequality for operator matrices, which improves an earlier inequality due to Hou and Du. As an application of this numerical radius inequality, we derive a new bound for the zeros of polynomials.     

Non-commutative operator Bohr inequality

It is shown that if A, B, X are Hilbert space operators such that X?γI, for the positive real number γ, and p,q>1 with 1/p+1/q=1, then |A−B|²?p|A|²+q|B|² with equality if and only if (1−p)A=B and γ||||A−B|²|||?|||p|A|²X+qX|B|²||| for every unitarily invariant norm. Moreover, if in addition A, B are normal and X is any Hilbert-Schmidt operator, then ‖δ_A,B²(X)‖₂?‖p|A|²X+qX|B|²‖₂ with equality if and only if (1−p)AX=XB.     

Norm inequalities for commutators of positive operators and applications

Let X, Y, and Z be operators on a Hilbert space such that X and Z are positive. It is shown that

Interpolated Young and Heinz inequalities

In this article, we interpolate the well-known Young’s inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. As an application of these interpolated versions, we study the monotonicity these interpolations obey.     

Jensen matrix inequalities and direct sums

Let A, X and Y be n-by-n complex matrices such that A is positive semi-definite and X, Y are contractions. We prove that if f is an increasing convex function on [0, ∞) such that f(0) ≤ 0, then the eigenvalues of f(|X*AY|) are dominated by those of X*f(A)X⊕Y* f(A)Y. Several related results are considered.     

Norm Inequalities for Commutators of Self-adjoint Operators

Let A, B, and X be bounded linear operators on a complex separable Hilbert space. It is shown that if A and B are self-adjoint with and for some real numbers a ₁, a ₂, b ₁, and b ₂, then for every unitarily invariant norm|||·|||,

复半正定矩阵的奇异值不等式

用Mn表示所有复矩阵组成的集合.对于A∈Mn,σ(A)=(σ1(A),…,σn(A)),其中σ1(A)≥…≥σn(A)是矩阵A的奇异值.本文给出证明:对于任意实数α,A,B∈Mn为半正定矩阵,优化不等式σ(A-|α|B) wlogσ(A+αB)成立,改进和推广了文[5]的结果.     

New sharp inequalities for operator means

New sharp multiplicative reverses of the operator means inequalities are presented, with a simple discussion of squaring an operator inequality. As a direct consequence, we extend the operator Pólya-Szegö inequality to arbitrary operator means. Furthermore, we obtain some new lower and upper bounds for the Tsallis relative operator entropy, operator monotone functions and positive linear maps.     

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.     

Generalized Bebiano-Lemos-Providência inequalities and their reverses

We improve Bebiano-Lemos-Providência inequality: For A,B?0