A complement of the Ando-Hiai inequality

In this paper, we present a complement of a generalized Ando-Hiai inequality due to Fujii and Kamei [M. Fujii, E. Kamei, Ando-Hiai inequality and Furuta inequality, Linear Algebra Appl. 416 (2006) 541-545]. Let A and B be positive operators on a Hilbert space H such that 0<m₁?A?M₁ and 0<m₂?B?M₂ for some scalars m_i?M_i (i=1,2), and let α∈[0,1]. Put for i=1,2. Then for each 0<r?1 and s?1

     

A difference counterpart to a matrix Hölder inequality

We point out a sharp reverse Cauchy-Schwarz/Hölder matrix inequality. The Cauchy-Schwarz version involves the usual matrix geometric mean: Let A_i and B_i be positive definite matrices such that 0<mA_i?B_i?MA_i for some scalars 0<m?M and i=1,2,?,n. Then

     

Singular values, norms, and commutators

Let and X_i, i=1,…,n, be bounded linear operators on a separable Hilbert space such that X_i is compact for i=1,…,n. It is shown that the singular values of are dominated by those of , where ‖·‖ is the usual operator norm. Among other applications of this inequality, we prove that if A and B are self-adjoint operators such that a₁?A?a₂ and b₁?B?b₂ for some real numbers and b₂, and if X is compact, then the singular values of the generalized commutator AX-XB are dominated by those of max(b₂-a₁,a₂-b₁)(X⊕X). This inequality proves a recent conjecture concerning the singular values of commutators. Several inequalities for norms of commutators are also given.     

Jensen’s inequality for operators without operator convexity

We give Jensen’s inequality for n-tuples of self-adjoint operators, unital n-tuples of positive linear mappings and real valued continuous convex functions with conditions on the bounds of the operators. We also study operator quasi-arithmetic means under the same conditions.     

A lenticular version of a von Neumann inequality

We generalize to lens-shaped domains the classical von Neumann inequality for the disk. Received: 29 March 2005; revised: 14 June 2005     

A Dunkl-Williams type inequality for absolute value operators

Dunkl and Williams showed that for any nonzero elements x,y in a normed linear space X

     

Rotfel’d type inequalities for norms

In this note, we consider some norm inequalities related to the Rotfel’d Trace Inequality

     

Generalized matrix version of reverse Hölder inequality

A generalized matrix version of reverse Cauchy-Schwarz/Hölder inequality is proved. This includes the recent results proved by Bourin, Fujii, Lee, Niezgoda and Seo.     

Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces

Some sharp bounds for the Euclidean operator radius of two bounded linear operators in Hilbert spaces are given. Their connection with Kittaneh’s recent results which provide sharp upper and lower bounds for the numerical radius of linear operators are also established.     

λ-Aluthge Iteration and Spectral Radius

Let (H) be an invertible operator on the complex Hilbert space H. For 0 < λ < 1, we extend Yamazaki’s formula of the spectral radius in terms of the λ-Aluthge transform where T = U|T| is the polar decomposition of T. Namely, we prove that where r(T) is the spectral radius of T and ||| · ||| is a unitarily invariant norm such that (B(H), ||| · |||) is a Banach algebra with ||| I ||| = 1. In memory of my brother-in-law, Johnny Kei-Sun Man, who passed away on January 16, 2008, at the age of fifty nine.     

A new proof of an inequality of Bohr for Hilbert space operators

In this note we present a new proof and an extension of the Hilbert space operators version of an inequality by Bohr.     

Estimates for norms of matrix-valued and operator-valued functions and some of their applications

A survey is presented of estimates for a norm of matrix-valued and operator-valued functions obtained by the author. These estimates improve the Gel'fand-Shilov estimate for regular functions of matrices and Carleman's estimates for resolvents of matrices and compact operators.From the estimates for resolvents, the well-known result for spectrum perturbations of self-adjoint operators is extended to quasi-Hermitian operators. In addition, the classical Schur and Brown's inequalities for eigenvalues of matrices are improved.From estimates for the exponential function (semigroups), bounds for solution norms of nonlinear differential equations are derived. These bounds give the stability criteria which make it possible to avoid the construction of Lyapunov functions in appropriate situations.     

Operator extensions of Hua’s inequality

We give an extension of Hua’s inequality in pre-Hilbert C^∗-modules without using convexity or the classical Hua’s inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert C^∗-modules, another extension of Hua’s inequality is obtained. We also present an operator Hua’s inequality, which is equivalent to operator convexity of given continuous real function.     

An asymmetric Kadison’s inequality

Some inequalities for positive linear maps on matrix algebras are given, especially asymmetric extensions of Kadison’s inequality and several operator versions of Chebyshev’s inequality. We also discuss well-known results around the matrix geometric mean and connect it with complex interpolation.     

Von Neumann’s inequality for noncommuting contractions

Let T₁,…,T_d be linear contractions on a complex Hilbert space and p a complex polynomial in d variables which is a sum of d single variable polynomials. We show that the operator norm of p(T₁,…,T_d) is bounded by

     

Square of ω-hyponormal operators

In this paper, we show that ifT is a -hyponormal operator, thenT ² is also -hyponormal.     

Inequalities for sums and direct sums of Hilbert space operators

We prove several singular value inequalities and norm inequalities involving sums and direct sums of Hilbert space operators. It is shown, among other inequalities, that if X and Y are compact operators, then the singular values of are dominated by those of X ⊕ Y. Applications of these inequalities are also given.     

Mourre's inequality and embedded bound states

A self-adjoint operator H with an eigenvalue λ embedded in the continuum spectrum is considered. Boundedness of all operators of the form AnP is proved, where P is the eigenprojection associated with λ and A is any self-adjoint operator satisfying Mourre's inequality in a neighborhood of λ and such that the higher commutators of H with A up to order n+2 are relatively bounded with respect to H.     

The simplest proof of Lieb concavity theorem

In this paper, by applying jointly concavity and jointly convexity of generalized perspective of some elementary functions, we give the simplest proof of the well-known Lieb concavity theorem and Ando convexity theorem.