Singular Value Inequalities for Compact Normal Operators

We give singular value inequality to compact normal operators, which states that if A is compact normal operator on a complex separable Hilbert space, where 1 2 A A iA   is the cartesian decomposition of A , then       1 2 1 2 12 j j j s A A s A s A A     for 1, j 2,   Moreover, we give inequality which asserts that if A is compact normal operator, then       1 2 j * 1 2 2 j j 2 s A s A A  A   s A iA   for 1 j , 2,   Several inequalities will be proved.

A A iA   is the cartesian decomposition of A , then  Moreover, we give inequality which asserts that if A is compact normal operator, then

Introduction
Let   B H denote the space of all bounded linear operators on a complex separable Hilbert space H, and let

 
of those of together with those of .Here, we use the direct sum notation for the blockdiagonal 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 are the positive operators given by 2 A A and 2 , see [1].

Let
be any operator, we can write in the form A A A A A A  .Audeh and Kittaneh have proved in [3] for 1, 2, j   Also, Audeh and Kittaneh have proved in [3] for 1, 2, j   In addition to this, Audeh and Kittaneh have proved in [3] 3) is a generalization of (1.4).
Hirzallah and Kittaneh have proved in [5] that if In this paper, we will give singular value inequalities for normal operators: Let be normal operator in for 1, 2, j   We will give singular value inequality to the normal operator * A iA  , where is normal:

Main Results
We will begin by presenting the following theorem for complex numbers Theorem 2.1.Let x a ib   be complex number.Then Proof.The right hand side of the inequalities is well known.To prove the left hand side, Now, we will present operator version of Theorem 2.1, inequality (2.1).

Theorem 2.2. Let be normal operator in
A A iA   be the Cartesian decomposition of the normal operator , which implies that  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   it fol- lows Weyl's monotonicity principle [1] and the inequality (2.4) that We will present operator version of Theorem 2.1, inequality (2.2).
Remark 2. Let 0 , 0 where is normal operator.Then is normal operator with , and . Now, by direct calculations and applying Theorem 2.2 we get for 1, 2, j   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 A 2 is less than or equal to the singular value of the normal operator itself.
As an application of the Theorem 2.
is normal operator, so we can write the Cartesian decomposition of as , and where the cartesian decomposition of is given by . By comparison of and we see easily that

A A i A A A A i A A A A iA iA A A iA iA T A
for 1, 2, j   We will give simple and new proof to the inequality apply the inequality (1.4) we get for 1, 2, j   Audeh and Kittaneh separates Jordan of self-adjoint operator in the inequality (1.3).Here we will give a shorter 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).

 
A K H  be self-adjoint operator.Then The following theorem is the second application of the inequality (1.5).

 
A K H  be self-adjoint operator.Then , so using the inequality (1.5) we get for 1, 2, j   For related Cauchy-Schwarz type inequalities, we refer to [2] and references therein.Here, we will present similar new inequality.