Fuad Kittaneh

On the singular values of a product of operators

For compact Hilbert space operators A and B, the singular values of

Notes on matrix arithmetic-geometric mean inequalities

A^ * B

Reverse Young and Heinz inequalities for matrices

are shown to be dominated by those of

Matrix Young inequalities for the Hilbert–Schmidt norm

\frac{1}{2}(AA^* + BB^* )

Norm inequalities for fractional powers of positive operators

.

A note on the arithmetic-geometric-mean inequality for matrices

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 matrix arithmetic–geometric mean inequalities and trace inequalities is discussed.

Norm Inequalities for Positive Operators

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.

Singular Values of Companion Matrices and Bounds on Zeros of Polynomials

Abstract Let A,B, and X be n×n complex matrices such that A and B are positive semidefinite. If p,q>1 with 1 p + 1 q =1 , it is shown that ∥ 1 p A p X+ 1 q XB q ∥ 2 2 ⩾ 1 r 2 A p X−XB q 2 2 + AXB 2 2 , where r=max(p,q) and · 2 is the Hilbert–Schmidt norm. Generalizations and applications of this inequality are also considered.

Clarkson Inequalities with Several 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 have |‖ArXBr|‖ ≤ |‖X|‖1-r|‖AXB|‖r. This is an extension of the classical Heinz-Kato inequality which was originally proved for the usual operator norm. Other related inequalities are also discussed.

On Lipschitz functions of normal operators

Abstract We give a simple proof of the inequality ⦀ AA ∗ X + XBB ∗ ⦀ ⩾ 2 ⦀ A ∗ AB ⦀, where A , B , and X are arbitrary n × n matrices and ⦀ · ⦀ is any unitarily invariant norm. For the Schatten p -norms ‖ · ‖ p with 1 p AA ∗ X + XBB ∗ ‖ p = 2‖ A ∗ XB ‖ p if and only if AA ∗ X = XBB ∗ .