An inequality for complex matrices (original) (raw)

Characterization of the class of unitary operators by operator inequalities

Linear and Multilinear Algebra, 2011

Let BðH Þ, IðH Þ and UðH Þ be the C Ã-algebra of all bounded linear operators acting on a complex Hilbert space H, the set of all invertible elements in BðH Þ and the class of all unitary operators in BðH Þ, respectively. In this note, we shall show that if S 2 IðH Þ, then the injective norm of S S À1 þ S À1 S in the tensor product space BðH Þ BðH Þ attains its minimal value 2 if and only if S is normal and satisfies the condition j þ j 2 for every , in the spectrum (S) of S. Finally, it is shown that if S 2 IðH Þ, then the inequality kSXS À1 þ S À1 XSk 2kXk holds for all X in BðH Þ if and only if S 2 R Ã UðH Þ.

On invertibility and positive invertibility of matrices

Linear Algebra and its Applications, 2001

New non-singularity and non-negative invertibility criteria for matrices are derived. They yield new bounds for the spectrum. Moreover, we obtain a bound for the norm of the inverse matrix and new estimates for the spectral radius.

On some operator norm inequalities

Linear Algebra and its Applications, 2004

Let B(H) be the C *-algebra of all bounded linear operators on a complex Hilbert space H, S be an invertible and selfadjoint operator in B(H) and let (I,. I) denote a norm ideal of B(H). In this note, we shall show the following inequality:

On operators satisfying an inequality

Journal of Inequalities and Applications, 2012

,B ∈ B(B(H)) denote either the generalized derivation δ A,B = L A -R B or the elementary operator A,B = L A R B -I, where L A and R B are the left and right multiplication operators defined on B(H) by L A = AX and R B = XB respectively. This article concerns some spectral properties of k-quasi- * -class A operators in a Hilbert space, as the property of being hereditarily polaroid. We also establish Weyl-type theorems for T and d A,B , where T is a k-quasi- * -class A operator and A, B * are also k-quasi- * -class A operators. MSC: Primary 47B47; 47A30; 47B20; secondary 47B10

The Drazin inverse and a spectral inequality of marcus, minc, and moyls

Journal of Optimization Theory and Applications, 1975

Some properties of the Drazin inverse of a square matrix are derived which, together with elementary divisor theory, are used to investigate a spectral inequality which relates the maximal eigenvalues of two nonnegative indecomposable matrices.

Matrix and operator inequalities

Journal of Inequalities in Pure and Applied Mathematics, 2001

In this paper we prove certain inequalities involving matrices and operators on Hilbert spaces. In particular inequalities involving the trace and the determinant of the product of certain positive definite matrices.

An operator inequality

Linear Algebra and its Applications, 1990

For Hilbert space operators, with S invertible hermitian, it is proved that IISTS-' + s-'TSII > 211Tll.