Advances in Linear Algebra & Matrix Theory
Vol.3 No.4(2013), Article ID:40525,7 pages DOI:10.4236/alamt.2013.34006
More Commutator Inequalities for Hilbert Space Operators
Department of Basic Sciences, Petra University, Amman, Jordan
Email: waudeh@uop.edu.jo
Copyright © 2013 Wasim Audeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received July 2, 2013; revised August 16, 2013; accepted September 3, 2013
Keywords: Compact Operator; Inequality; Positive Operator; Self-Adjoint Operator; Singular Value
ABSTRACT
The well-known arithmetic-geometric mean inequality for singular values, according to Bhatia and Kittaneh, says that if and
are compact operators on a complex separable Hilbert space, then
Hirzallah has proved that if
are compact operators, then
We give inequality which is equivalent to and more general than the above inequalities, which states that if
are compact operators, then
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.
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