More Results on Singular Value Inequalities for Compact Operators ()
1. Introduction
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
(1.1)
Hirzallah has proved in [4] that if 
, then
(1.2)
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
(1.3)
Audeh and Kittaneh have proved in [5] that if 
such that 
is self-adjoint, 
, then
(1.4)
On the other hand, Tao has proved in [6]
that if 
such that
, then
(1.5)
Moreover, Zhan has proved in [7] that if 
are positive, then
(1.6)
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
(1.7)
Bhatia and Kittaneh have proved in [8] that if
, such that 
is self-adjoint, 
, and
, then
(1.8)
Audeh and Kittaneh have proved in [5]
that if 
such that
, then
(1.9)
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
(1.10)
2. Main Result
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.