Inequalities for some operator and matrix functions

Matrix and Operator Inequalities

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.

Some weighted operator geometric mean inequalities

In this paper, using the extended Holder- -McCarthy inequality, several inequalities involving the α-weighted geometric mean (0<α<1) of two positive operators are established. In particular, it is proved that if A,B,X,Y∈B(H) such that A and B are two positive invertible operators, then for all r ≥1, ‖X^* (A⋕_α B)Y‖^r≤‖〖(X〗^* AX)^r ‖^((1-α)/2) ‖〖(Y〗^* AY)^r ‖^((1-α)/2) ‖〖(X〗^* BX)^r ‖^(α/2) ‖〖(Y...

From Matrix to Operator Inequalities

We generalize Löwner’s method for proving that matrix monotone functions are operator monotone. The relation x ≤ y on bounded operators is our model for a definition for C∗-relations of being residually finite dimensional. Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved, and veri...

Some Inequalities for Reliability Functions

On some matrix inequalities

The arithmetic-geometric mean inequality for singular values due to Bhatia and Kittaneh says that 2sj(AB ∗) ≤ sj(A∗A + B∗B), j = 1, 2, . . . for any matrices A,B. We first give new proofs of this inequality and its equivalent form. Then we use it to prove the following trace inequality: Let A0 be a positive definite matrix and A1, . . . , Ak be positive semidefinite matrices. Then tr k ∑