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![](https://www.scirp.org/html/1-2230024\23289ccf-fe0b-404e-972a-671cfddb158a.jpg)
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)
![](https://www.scirp.org/html/1-2230024\657bfbb0-205e-4d97-a935-867c65a6b8ef.jpg)
Hirzallah has proved in [4] that if
, then
(1.2)
![](https://www.scirp.org/html/1-2230024\4cf89abe-40b3-4805-b7da-3c4dd606a5e1.jpg)
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)
![](https://www.scirp.org/html/1-2230024\ff98dbcf-9c4a-4c18-93f2-a8bd7d43d47f.jpg)
Audeh and Kittaneh have proved in [5] that if
such that
is self-adjoint, ![](https://www.scirp.org/html/1-2230024\b0fdf877-2700-4201-b339-bfa2bbf76680.jpg)
, 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)
![](https://www.scirp.org/html/1-2230024\93bbac12-3f0f-4f15-a924-2d034f549513.jpg)
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
![](https://www.scirp.org/html/1-2230024\178e1600-97a9-41fa-a88f-27190955d454.jpg)
![](https://www.scirp.org/html/1-2230024\1a1b06fd-380c-47d0-b75d-7db66c0e179f.jpg)
Proof. Let
,
Then
, and
![](https://www.scirp.org/html/1-2230024\7098be3a-58f5-4af2-918c-71fac3c3bf6e.jpg)
Now, using (1.1) we get
![](https://www.scirp.org/html/1-2230024\633b9107-3ea8-47bd-ade2-3616dc9871a9.jpg)
![](https://www.scirp.org/html/1-2230024\fb30ae75-8bf9-4245-8cb0-b4f849013d86.jpg)
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
![](https://www.scirp.org/html/1-2230024\6359a6ed-7f9e-4333-870a-1039641de07e.jpg)
![](https://www.scirp.org/html/1-2230024\e665bcd7-9c3c-4aa5-a1d1-f0108b53adcb.jpg)
Now, we prove that the inequalities (1.1) and (1.3) are equivalent.
Theorem 2.2. The following statements are equivalent:
(i) If
, then ![](https://www.scirp.org/html/1-2230024\aa6c9ccf-529b-43e5-8503-21979af3fa2d.jpg)
![](https://www.scirp.org/html/1-2230024\6fcddb2e-091e-483a-a99b-114458e2b24e.jpg)
(ii) Let
Then
![](https://www.scirp.org/html/1-2230024\119f5ee9-5509-4ac5-8bb9-2d79f9fcae01.jpg)
![](https://www.scirp.org/html/1-2230024\f08065d6-4ebe-48c9-a7bc-b363ba609b35.jpg)
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
![](https://www.scirp.org/html/1-2230024\49bc847d-f178-4beb-847b-36674229f591.jpg)
Proof. Since
it follows that
![](https://www.scirp.org/html/1-2230024\514dde67-d2a0-4892-a89d-3b87d09f5258.jpg)
In fact, if
then
is unitary and
![](https://www.scirp.org/html/1-2230024\e494450b-de30-4466-97f9-25057342c634.jpg)
Thus
![](https://www.scirp.org/html/1-2230024\ae026450-e8da-40d2-91a0-a7c458d951b1.jpg)
and so by applying the inequality (1.4), we get
![](https://www.scirp.org/html/1-2230024\052de305-64d3-4c8e-9ad1-1edf80a01e9b.jpg)
This is equivalent to saying that
![](https://www.scirp.org/html/1-2230024\10750d81-8a08-43df-930b-1fc8465f5c92.jpg)
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
![](https://www.scirp.org/html/1-2230024\94f86348-6fd6-4674-b3bb-75e865a75b86.jpg)
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
![](https://www.scirp.org/html/1-2230024\f6a01222-3d84-433e-adf0-96b61c3a9691.jpg)
![](https://www.scirp.org/html/1-2230024\c60552e0-e2f6-46f6-ba2c-6bc60c236502.jpg)
(ii) Let
such that
![](https://www.scirp.org/html/1-2230024\74403e7a-b8f4-4450-90a7-a0374e4b9bff.jpg)
Then ![](https://www.scirp.org/html/1-2230024\70611897-aec3-4759-b5e0-60e3ea105b65.jpg)
![](https://www.scirp.org/html/1-2230024\681d45ac-a449-4409-9c26-35b562721c21.jpg)
Proof.
This implication follows from the proof of Theorem 2.3.
Let
such that
is selfadjoint, ![](https://www.scirp.org/html/1-2230024\0b5de87c-96eb-4a6b-9029-a1a38e90b34a.jpg)
Then the matrix
![](https://www.scirp.org/html/1-2230024\223765a0-a955-4c53-af87-234046059554.jpg)
In fact, if
then
is unitary and
![](https://www.scirp.org/html/1-2230024\143d5426-4aa9-49ba-a683-99276c0f90c8.jpg)
Thus, by applying (ii) we get
![](https://www.scirp.org/html/1-2230024\1135c4ef-eb36-4588-9ad8-dd55a6ae6c41.jpg)
![](https://www.scirp.org/html/1-2230024\39af0c75-fc39-4e49-a8aa-9ebbce18b8b6.jpg)
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
![](https://www.scirp.org/html/1-2230024\e2f71797-752e-444d-ae9c-034d41b7819f.jpg)
![](https://www.scirp.org/html/1-2230024\ead93c6f-f5e0-4d03-8b41-6ac08c674a83.jpg)
Proof. As in the proof of Theorem 2.3., we have
![](https://www.scirp.org/html/1-2230024\b91ad0c7-ca54-4b4e-b5f2-6fdbe0615485.jpg)
and so by applying the inequality (1.8), we get
![](https://www.scirp.org/html/1-2230024\bd657d78-825a-459c-bc08-cc95cabf9626.jpg)
This is equivalent to saying that
![](https://www.scirp.org/html/1-2230024\b9f5b702-eff0-4746-b6b0-853a05e5cf3c.jpg)
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
![](https://www.scirp.org/html/1-2230024\aa592c61-7d66-446f-9eff-f57b9a2f7c55.jpg)
Following Weyl’s monotonicity principle, we have
![](https://www.scirp.org/html/1-2230024\c3642f40-8b60-4394-aac7-0974be945198.jpg)
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
![](https://www.scirp.org/html/1-2230024\7eabd52f-8aa9-49a6-bd36-09284fa35970.jpg)
(ii) ![](https://www.scirp.org/html/1-2230024\62920e7b-c905-4e0a-901f-004afe1c0cab.jpg)
(iii) Let
such that
Then
![](https://www.scirp.org/html/1-2230024\eb3fb19c-41ca-4d9e-ac33-9a248d356efb.jpg)
![](https://www.scirp.org/html/1-2230024\2f2abbb0-5d5e-4d88-bd76-83f6e32c510a.jpg)
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
![](https://www.scirp.org/html/1-2230024\354243a7-6616-4e85-9a84-4aed16199a8d.jpg)
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.