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 
and 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 
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 Jordan decomposition for self-adjoint operators asserts that every self-adjoint operator can be expressed as the difference of two positive operators. In fact, if 
is self-adjoint, then 
where
are the positive operators given by
and
, see [1].
Let 
be any operator, we can write 
in the form
, where 
and 
are self-adjoint operators, this is called the Cartesian decomposition of the operator
. If 
is normal, then
.
Audeh and Kittaneh have proved in [3] that if 
such that
, then
(1.1)
Also, Audeh and Kittaneh have proved in [3] that if 
such that 
is self-adjoint, 
, then
(1.2)
In addition to this, Audeh and Kittaneh have proved in [3] that if 
be self-adjoint operators, then
(1.3)
Zhan has proved in [4] that if 
are positive, then
(1.4)
Moreover, it has proved in [3] that (1.3) is a generalization of (1.4).
Hirzallah and Kittaneh have proved in [5] that if
, then
(1.5)
In this paper, we will give singular value inequalities for normal operators:
Let 
be normal operator in
. Then
(1.6)
We will give singular value inequality to the normal operator
, where 
is normal:
Let 
be normal operator in
. Then
(1.7)
2. Main Results
We will begin by presenting the following theorem for complex numbers Theorem 2.1. Let 
be complex number. Then
(2.1)
Also,
(2.2)
Proof. The right hand side of the inequalities is well known. To prove the left hand side,
Moreover, 
Now, we will present operator version of Theorem 2.1, inequality (2.1).
Theorem 2.2. Let 
be normal operator in
, where 
be the Cartesian decomposition of
. Then
Proof. Let 
be the Cartesian decomposition of the normal operator
, which implies that
. Now, 
, it follows that
In fact 
for
By using Weyl’s monotonicity principle [1]
and the inequality
, we get the right hand side of the theorem. To prove the left hand side of the inequality, we will use the inequality which is well known for commuting self-adjoint operators and it asserts that
(2.3)
This implies that
(2.4)
But it is known that
it follows Weyl’s monotonicity principle [1] and the inequality (2.4) that
(2.5)
Inequality (2.5) is equivalent to saying that
Remark 1. (i) Equality holds in the right hand side of Theorem 2.2 if either 
or
.
(ii) Equality holds in the left hand side of theorem 2.2 if
.
We will present operator version of Theorem 2.1, inequality (2.2).
Remark 2. Let
where 
is normal operator. Then 
is normal operator with 
is the Cartesian decomposition of
.
and
.
It follows that
, and
.
Now, by direct calculations and applying Theorem 2.2 we get
(2.6)
Remark 3. We note that the right hand side of the inequality (2.6) is the same as the inequality (1.6), but the left hand side of the inequalities (1.6) and (2.6) says that the singular value of the addition or subtraction of the Cartesian decomposition for the normal operator 
divided by 
is less than or equal to the singular value of the normal operator itself.
As an application of the Theorem 2.2, we will determine upper and lower bounds for singular values of the normal operator
, where 
is normal.
Theorem 2.3. Let 
be normal operator, where 
is the Cartesian decomposition of
. Then
Proof. Note that 
is normal operator, so we can write the Cartesian decomposition of 
as
where
, and
where the cartesian decomposition of 
is given by
. By making comparison of 
and 
we see easily that
. It follows that 
. Moreover,
Similarly,
. Now, apply Theorem 2.2 to get
(2.7)
This is equivalent to saying that
We will give simple and new proof to the inequality
(1.2).
Theorem 2.4. Let 
such that 
is self-adjoint, 
, then
Proof. Since 
is self-adjoint operator, we can write
in the form 
apply the inequality (1.4) we get
which is equivalent to saying that
Audeh and Kittaneh separates Jordan of self-adjoint operator in the inequality (1.3). Here we will give a shorter proof.
Theorem 2.5. Let 
be self-adjoint operators. Then
Proof. Since 
and 
are self-adjoint operators, we can write 
in the form 
and similarly we will write 
in the form
. Apply the inequality (1.4) we get
We will present the following two theorems as an application to the inequality (1.5).
Theorem 2.6. Let 
be self-adjoint operator. Then
(2.8)
Proof. It was proved in Theorem 2.2 that if 
is normal operator with Cartesian decomposition 
, then 
from this, it follows that
The following theorem is the second application of the inequality (1.5).
Theorem 2.7. Let 
be self-adjoint operator. Then
(2.9)
Moreover,
(2.10)
Proof. It is well known that
, so using the inequality (1.5) we get
Similarly, 
so using the inequality (1.5) we get
Bhatia and Kittaneh have proved in [6] that if
, then
For related Cauchy-Schwarz type inequalities, we refer to [2] and references therein. Here, we will present similar new inequality.
Theorem 2.8. Let 
be operators. Then
(2.11)
Proof. Suppose 
and 
This implies that
, and 
On the other hand, we have
, and
.
Since 
and 
are positive operators, then
is positive operator. Now by applying the inequality (1.1), we get
