Let denote the space of all bounded linear operators on a complex separable Hilbert space H, and let denote the two-sided ideal of compact operators in. For, the singular values of, denoted by are the eigenvalues of the positive operator as

repeated according to multiplicity. Note that It follows Weyl’s monotonicity principle (see, e.g., [1, p. 63] or [2, p. 26]) that if are positive and, then Moreover, for, if and only if The singular values of and are the same, and they consist of those of together with those of. Here, we use the direct sum notation for the blockdiagonal operator defined on.

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

Hirzallah has proved in [4] that if , then

In this paper, we will give a new inequality which is equivalent to and more general than the inequalities (1.1) and (1.2):

If, then

Audeh and Kittaneh have proved in [5] that if such that is self-adjoint, , then

On the other hand, Tao has proved in [6]

that if such that, then

Moreover, Zhan has proved in [7] that if are positive, then

We will give a new inequality which generalizes (1.5), and is equivalent to the inequalities (1.1), (1.2), (1.3), (1.4), (1.5), and (1.6):

Let such that

, then

Bhatia and Kittaneh have proved in [8] that if, such that is self-adjoint, , and, then

Audeh and Kittaneh have proved in [5]

that if such that, then

We will prove a new inequality which generalizes (1.9), and is equivalent to the inequalities (1.8) and (1.9):

If such that

, then

Our first singular value inequality is equivalent to and more general than the inequalities (1.1) and (1.2).

Theorem 2.1 Let Then

Proof. Let, Then

, and

Now, using (1.1) we get

Remark 1. As a special case of (1.3), let .we get (1.1)

Remark 2. As a special case of (1.3), let we get (1.2), to see this:

Replace we get

Now, we prove that the inequalities (1.1) and (1.3) are equivalent.

Theorem 2.2. The following statements are equivalent:

(i) If, then

(ii) Let Then

Proof. 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. Chaining this with results in [5], we get that the inequalities (1.1), (1.2), (1.3), (1.4), (1.5), and (1.6) are equivalent.

Our second singular value inequality is equivalent to the inequality (1.4).

Theorem 2.3. Let such that

Then

Proof. Since

it follows that

In fact, if then is unitary and

Thus

and so by applying the inequality (1.4), we get

This is equivalent to saying that

Remark 4. 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:

(i) Let such that is self-adjoint, Then

(ii) Let such that

Then

Proof. This implication follows from the proof of Theorem 2.3.

Let such that is selfadjoint, Then the matrix

In fact, if then is unitary and

Thus, by applying (ii) we get

Remark 5. From equivalence of inequalities (1.4) and (1.7) in Theorem 2.4, and equivalence of the inequalities (1.1), (1.2), (1.3), (1.4), (1.5), and (1.6) in Remark 3, we get that the inequalities (1.1), (1.2), (1.3), (1.4), (1.5), (1.6) and (1.7) are equivalent.

Our third singular value inequality is equivalent to the inequalities (1.8) and (1.9).

Theorem 2.5. Let such that

Then

Proof. As in the proof of Theorem 2.3., we have

and so by applying the inequality (1.8), we get

This is equivalent to saying that

Remark 6. 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:

If  Then

and so

Following Weyl’s monotonicity principle, we have

Chaining this with the inequality (1.7), yields the inequality (1.10).

Now, we prove that the inequalities (1.8) and (1.10) are equivalent.

Theorem 2.6. The following statements are equivalent:

(i)         Let, such that is self-adjoint, , and, then

(ii)      

(iii)      Let such that

Then

Proof. This implication follows the proof of Theorem 2.5.

As in the proof of Theorem 2.4, if is self-adjoint, Then

.

Thus, by (ii) we have

Remark 7. From equivalence of inequalities (1.8) and (1.10) in Theorem 2.6, and equivalence of inequalities (1.8) and (1.9) in [5], we get that the inequalities (1.8), (1.9), and (1.10) are equivalent.