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The well-known arithmetic-geometric mean inequality for singular values, due to Bhatia and Kittaneh, is one of the most important singular value inequalities for compact operators. The purpose of this study is to give new singular value inequalities for compact operators and prove that these inequalities are equivalent to arithmetic-geometric mean inequality, the way by which several future studies could be done.

Let

S together with those of T.

Bhatia and Kittaneh have proved in [

for

Audeh and Kittaneh in [

If

for

The well-known arithmetic-geometric mean inequality for singular values, due to Bhatia and Kittaneh [

for

for

that

for

Audeh and Kittaneh have proved in [

If

for

It has been pointed out in [

Moreover, Tao in [

for

In this study, we will present several new inequalities, and prove that they are equivalent to arithmetic-geometric mean inequality.

The following are the proved inequalities in this study:

Let

for

Let

for

Let

for

If

for

Let

for

Our first singular value inequality needs the following lemma.

Lemma 1: Let

Now we will prove the first Theorem which is equivalent to arithmetic- geometric mean inequality.

Theorem 3.1 Let

for

Proof. Let

From (1.5) we have

for

Now we will prove that Theorem (3.1) is equivalent to arithmetic-geometric mean inequality.

Theorem 3.2 The following statements are equivalent:

1) Let

for

2) Let

for

Proof. 1) ® 2) Let

Now apply arithmetic-geometric mean inequality to get

for

The above steps implies that

2) ® 1) The matrix

for

for

for

The following lemma which was proved by Bhatia [

Lemma 2 Let

Now we will prove the following theorem which is more general than Theo- rem (3.1) and equivalent to arithmetic-geometric mean inequality.

Theorem 3.3 Let

for

Proof. Applying Lemma (2) gives

Remark 1 Theorem (3.3) is generalization of Theorem (3.1) because here X is arbitrary operator but there A should be positive operator.

Remark 2 Inequality (2.2) is equivalent to arithmetic-geometric mean inequality. We can prove this equivalent by similar steps used to prove Theorem (3.2).

The following theorem is a generalization of Theorem (3.1) and Theorem (3.3).

Theorem 3.4 Let

for

Proof. Let

use Inequality (1.5) to get the required result.

Remark 3 Replace B, D by 0 in Inequality (2.4) will gives Inequality (2.1).

Remark 4 Replace A, C by 0 in Inequality (2.4) will also gives Inequality (2.1).

Now we will use Inequality (1.3) to prove the following theorem, then we will show that they are equivalent.

Theorem 3.5 Let

for

Proof. Let

for

Now we will prove that Inequality (2.3) is equivalent to Inequality (1.3).

Theorem 3.6 The following statements are equivalent:

1) Let

for

2) Let

for

Proof. 1) ® 2) It is the proof of Theorem (3.5).

2) ® 1) By replacing

get

In the rest of this paper, we will prove new inequality which is equivalent to Inequality (1.7).

Theorem 3.7 Let

for

Proof. Let

and

Inequality (1.7) we get the result.

We will prove that Inequality (1.7) is equivalent to Inequality (3.5).

Theorem 3.8 The following statements are equivalent:

1) Let

for

2) Let

for

Proof. 1) ® 2) This implication follows from the proof of Theorem 3.7.

2) ® 1) Let

for

If and only if

for

for

Since this study has been completed, we can conclude that several singular value inequalities for compact operators are equivalent to arithmetic-geometric mean inequality, which in turns have many crucial applications in operator theory, and from this point we advise interested authors to join these results with results in other studies to make connection between several branches in operator theory.

The author is grateful to the University of Petra for its Support. The Author is grateful to the referee for his comments and suggestions.

Audeh, W. (2017) Applications of Arithmetic Geometric Mean Inequality. Advances in Linear Algebra & Matrix Theory, 7, 29-36. https://doi.org/10.4236/alamt.2017.72004