1. Fundamental Principles
Let
indicate the set of all bounded linear operators on a complex separable Hilbert space H, and let
indicate the two-sided ideal of compact operators in
. If
, the singular values of T, denoted by
are the eigenvalues of the positive operator
ordered as
and repeated according to multiplicity. It is well known that
for
. It follows by Weyl’s monotonicity principle (see, e.g., [1] , p. 63 or [2] , p. 26) that if
are positive and
, then
for
. Moreover, for
if and only if
for
. Here, we use the direct sum notation
for the block-diagonal operator
defined on
. The sin- gular values of
and
are the same, and they consist of those of
S together with those of T.
Bhatia and Kittaneh have proved in [3] that if
such that
is self-adjoint,
, and
, then
(1.1)
for
.
Audeh and Kittaneh in [4] prove inequality which is equivalent to inequality (1.1):
If
such that
, then
(1.2)
for
.
The well-known arithmetic-geometric mean inequality for singular values, due to Bhatia and Kittaneh [5] , says that if
, then
(1.3)
for
. On the other hand, Zhan has proved in [6] that if
are positive, then
(1.4)
for
. Moreover, Tao has proved in [7] that if
such
that
, then
(1.5)
for
.
Audeh and Kittaneh have proved in [4] that:
If
such that
is self-adjoint,
, and
, then
(1.6)
for
.
It has been pointed out in [4] that the four inequalities (1.3)-(1.6) are equi- valent.
Moreover, Tao in [7] uses inequality (1.3) to prove that if
and
are positive operators in
,
. Then
(1.7)
for
.
2. Introduction
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
and
be operators in
where
,
and
arbitrary operators. Then
(2.1)
for
.
Let
and
be arbitrary operators in
. Then we have
(2.2)
for
.
Let
be operators in
. Then
(2.3)
for
.
If
and
are operators in
. Then
(2.4)
for
.
Let
be positive operators in
Then
(2.5)
for
.
3. Main Results
Our first singular value inequality needs the following lemma.
Lemma 1: Let
be a positive operator in
,
be an arbitrary operator in
. Then we have
(3.1)
Now we will prove the first Theorem which is equivalent to arithmetic- geometric mean inequality.
Theorem 3.1 Let
and
be operators in
where
,
and
arbitrary operators. Then
![]()
for
.
Proof. Let
(because
by assumption), and let
. Then we have
![]()
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
, then
![]()
for
.
2) Let
and
be operators in
where
,
and
arbitrary operators. Then
![]()
for
.
Proof. 1) ® 2) Let ![]()
Now apply arithmetic-geometric mean inequality to get
![]()
for
. But
![]()
The above steps implies that
for
.
2) ® 1) The matrix
can be factorized as
, but it is well known that
for
. So
![]()
for
, from (2) we have
(3.2)
for
. Now let
in Inequality (3.2) we get
(3.3)
for
, which is the arithmetic-geometric mean inequality.
The following lemma which was proved by Bhatia [1] is essential to prove the next theorem.
Lemma 2 Let
be arbitrary operator in
. Then
(3.4)
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
and
be arbitrary operators in
. Then we have
![]()
for
.
Proof. Applying Lemma (2) gives
for an arbitrary ope- rator
. Let
by using Inequality (3.1) we have
Hence using Inequality (1.5) 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
and
be arbitrary operators in
. Then we have
![]()
for
.
Proof. Let
Then
Hence
![]()
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
be operators in
. Then
![]()
for
.
Proof. Let
Then
and
Now use Inequality (1.3) we get
![]()
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
. Then
![]()
for
.
2) Let
be operators in
. Then
![]()
for
.
Proof. 1) ® 2) It is the proof of Theorem (3.5).
2) ® 1) By replacing
and
in Inequality (2.3), we
get
From this we reach to
which implies that
for
.
In the rest of this paper, we will prove new inequality which is equivalent to Inequality (1.7).
Theorem 3.7 Let
be positive operators in
, n is an even integer,
. Then
(3.5)
for
.
Proof. Let
Then we have
![]()
and
Now apply
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
and
be positive operators in
,
. Then
![]()
for
.
2) Let
be positive operators in
, n is even integer,
. Then
![]()
for
.
Proof. 1) ® 2) This implication follows from the proof of Theorem 3.7.
2) ® 1) Let
in Inequality (3.5) to get
![]()
for
. But
and
for
.
If and only if
, this gives
![]()
for
, replace
by
,
by
in this inequality we will get
![]()
for
.
4. Conclusion
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.
Acknowledgements
The author is grateful to the University of Petra for its Support. The Author is grateful to the referee for his comments and suggestions.