More Results on Singular Value Inequalities for Compact Operators

The well-known arithmetic-geometric mean inequality for singular values, according to Bhatia and Kittaneh, says that if A and B are compact operators on a complex separable Hilbert space, then     * * * 2 j j s AB s A A B B   for 1, 2, j   Hirzallah has proved that if 1 2 3 4 , , , and A A A A are compact operators, then 1 1 3 * * 2 1 2 3 4 2 4 2 j j A A s A A A A s A A                     for 1, 2, j   We give inequality which is equivalent to and more general than the above inequalities, which states that if , , , 1, 2, , i i A B i n   are compact operators, then

 We give inequality which is equivalent to and more general than the above inequalities, which states that if , , , 0 0 0 0 0 0 2 0 0 0 0 0 0

Introduction
Let   B H denote the space of all bounded linear operators on a complex separable Hilbert space H, and let

 
The well-known arithmetic-geometric mean inequality for singular values, according to Bhatia and Kittaneh [3], says that if

, , , and
for 1, 2, j   In this paper, we will give a new inequality which is equivalent to and more general than the inequalities (1.1) and (1.2): ,  On the other hand, Tao has proved in [6] that if , , , ,  Audeh and Kittaneh have proved in [5] that if  We will prove a new inequality which generalizes (1.9), and is equivalent to the inequalities (1.8) and (1.9): , , , ,

Main Result
Our first singular value inequality is equivalent to and more general than the inequalities (1.1) and (1.2).
 Now, we prove that the inequalities (1.1) and (1.3) are equivalent.
This implication follows from the proof of Theorem 2.1.
This implication follows from Remark 1. Remark 3. It can be shown trivially that (1.1) and (1.2) are equivalent.By using this with Theorem 2.2, we conclude that the inequalities (1.2) and (1.3) are equivalent.
Our second singular value inequality is equivalent to the inequality (1.4).
, , , , then U is unitary and and so by applying the inequality (1.4), we get While the proof of the inequality (1.7), given in Theorem 2.3 is based on the inequality (1.4), it can be obtained by applying the inequality (1.6) to the positive operators Now, we prove that the inequalities (1.4) and (1.7) are equivalent.
Theorem 2.4.The following statements are equivalent: , , , , This implication follows from the proof of Theorem 2.3.
Our third singular value inequality is equivalent to the inequalities (1.8) and (1.9).
, , , , As in the proof of Theorem 2.3., we have and so by applying the inequality (1.8), we get While the proof of the inequality (1.10), given in Theorem 2.5 is based on the inequality (1.8), it can be obtained by employing the inequality (1.7) as follows: and so  , , , , This implication follows the proof of Theorem 2.5.

K
H denote the two-sided ideal of compact operators in