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An extension of the Golden-Thompson theorem
Journal of Inequalities and Applications volume 2014, Article number: 14 (2014)
In this paper, we shall prove for normal matrices A, B. In particular, if A, B are Hermitian matrices, yielding the Golden-Thompson inequality.
MSC:15A16, 47A63, 15A45.
1 Introduction and preliminaries
The famous Golden-Thompson inequality [1–4] for Hermitian matrices A, B states that . This inequality is a basic tool in quantum statistical mechanics and extensions to infinite dimension have an extensive literature [5, 6]. In this paper, we extend the classical Golden-Thompson theorem to normal matrices.
Throughout this paper, we adopt the following notation. Let be the set of all complex matrices. For a matrix , as usual, its conjugate transpose is denoted by . A matrix A is called Hermitian if , normal if , and unitary if ( is the identity matrix of order n). Given a matrix , the eigenvalues and singular values of A are denoted by , and , respectively, where and . In particular, when A is positive semidefinite (), then . For simplicity, we denote and . Recall that the singular values of a matrix are defined to be the eigenvalues of , i.e., . Here is the spectral norm of A. It is known that the spectral norm over is unitarily invariant, i.e., for all unitary matrices U, V.
We now recall the concept of majorization (details can be found in [7–9]). We have the following basic majorant relations. For real vectors , in coordinates in decreasing order, we say that x is weakly majorized by y, denoted by , if
and the weak log-majorant relation means
Remark 1.1 For , we denote . Weyl’s majorant theorem  says that for , that is,
The formula above implies that .
In this section, we shall propose some lemmas, laying the foundations of our main results in the next section.
Lemma 2.1 
If A, B are positive semidefinite matrices, then
Here note that is the spectral norm of X.
Lemma 2.2 If are normal matrices, then for any integer
Since the norm is unitary invariant, we obtain the following:
From Lemma 2.1, , we therefore conclude that
Lemma 2.3 If are normal matrices, then
Proof Since A and B are normal, it follows from Lemma 2.2 that . So we get
as desired. □
Lemma 2.4 If are normal matrices, then for integers
Proof By Lemma 2.3, we have , and
Applying Lemma 2.1 to the right side above, we have the following:
Thus we get for integers , as desired. □
Here we note that holds for any normal matrix.
The following lemma needs the notion of the Grassmann power (or antisymmetric tensor product), which can be found in [, p.18].
Lemma 2.5 If , , then for any natural number m, the following holds:
Proof For , consider the k th antisymmetric tensor product of . It is known [, p.18] that and
which, equivalently, says that . This completes the proof. □
3 Main results
In this section, we shall present the main results of this paper.
Theorem 3.1 If are normal matrices, then
Proof Let be normal matrices. It is clear that , are normal for . By replacing A, B by , in Lemma 2.4, respectively, we can obtain the following for integers :
Here we note that because . So we obtain
From Lemma 2.5, we have
The Lie product formula [, p.254] says that for any matrices A, B
Thus taking in the inequality above yields
Finally we note that
Thus we get
This completes the proof. □
From Theorem 3.1, we know that
On the other hand, the following equation holds:
The above two inequalities yield the following:
Thus, we can get the following corollary by using the Fan Dominance Principle [, p.56].
Corollary 3.2 If are normal matrices, then
for all unitarily invariant norms .
From Theorem 3.1, we can also have the following result.
Theorem 3.3 If are normal matrices, then
Proof By Weyl’s majorant theorem we have . Hence Theorem 3.1 implies the desired inequality in Theorem 3.3. □
Note that Theorem 3.3 strengthens the Golden-Thompson inequality:
for Hermitian matrices A, B.
Theorem 3.4 If A, B are normal matrices, then
Proof Because implies , it follows from Theorem 3.3 that
Taking the traces above, we have
So we get the desired inequality. This completes the proof. □
Of course, Theorem 3.4 is an extension of Golden-Thompson inequality:
for Hermitian matrices A, B.
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The authors would like to thank the referees for reading this work carefully, providing valuable suggestions and comments, which have significantly improved this article. This work is supported by National Natural Science Foundation of China (Grant No. 61379001).
The authors declare that they have no competing interests.
HL carried out the theorems and corresponding proofs, DZ checked the proofs carefully, and provided numerical examples and valuable suggestions. All authors read and approved the final manuscript.
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Li, H., Zhao, D. An extension of the Golden-Thompson theorem. J Inequal Appl 2014, 14 (2014). https://doi.org/10.1186/1029-242X-2014-14
- normal matrix
- Golden-Thompson inequality