New Refinement of the Operator Kantorovich Inequality

We focus on the improvement of operator Kantorovich type inequalities. Among the consequences, we improve the main result of the paper [H.R. Moradi, I.H. Gümüş, Z. Heydarbeygi, A glimpse at the operator Kantorovich inequality, Linear Multilinear Algebra, doi:10.1080/03081087.2018.1441799].

Extension of the Kantorovich inequality for positive multilinear mappings

It is known that the power function f (t) = t2 is not matrix monotone. Recently, it has been shown that t2 preserves the order in some matrix inequalities. We prove that if A = (A1,...,Ak) and B = (B1,...,Bk) are k-tuples of positive matrices with 0 < m ? Ai; Bi ? M (i = 1,...,k) for some positive real numbers m < M, then ?2 (A-11,...,A-1k) ? (1+vk)2/4vk)2 ?-2(A1,...,Ak) and ?2 (A1+B1/2,..., Ak+Bk/2)? (1+vk)2/4vk)2 ?2 (A1#B1,...Ak#Bk), where ? is a unital positive multilinear mapping and v = M/m is the condition number of each Ai.

Refinements of Kantorovich Inequality for Hermitian Matrices

Some new Kantorovich-type inequalities for Hermitian matrix are proposed in this paper. We consider what happens to these inequalities when the positive definite matrix is allowed to be invertible and provides refinements of the classical results.

Approximations of Antieigenvalue and Antieigenvalue-Type Quantities

We will extend the definition of antieigenvalue of an operator to antieigenvalue-type quantities, in the first section of this paper, in such a way that the relations between antieigenvalue-type quantities and their corresponding Kantorovich-type inequalities are analogous to those of antieigenvalue and Kantorovich inequality. In the second section, we approximate several antieigenvalue-type quantities for arbitrary accretive operators. Each antieigenvalue-type quantity is approximated in terms of the same quantity for normal matrices. In particular, we show that for an arbitrary accretive operator, each antieigenvalue-type quantity is the limit of the same quantity for a sequence of finite-dimensional normal matrices.