An extension of the Golden-Thompson theorem
© Li and Zhao; licensee Springer. 2014
Received: 5 July 2013
Accepted: 12 December 2013
Published: 9 January 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.
Keywordsnormal matrix majorization Golden-Thompson inequality
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.
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 
Here note that is the spectral norm of X.
as desired. □
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].
which, equivalently, says that . This completes the proof. □
3 Main results
In this section, we shall present the main results of this paper.
This completes the proof. □
Thus, we can get the following corollary by using the Fan Dominance Principle [, p.56].
for all unitarily invariant norms .
From Theorem 3.1, we can also have the following result.
Proof By Weyl’s majorant theorem we have . Hence Theorem 3.1 implies the desired inequality in Theorem 3.3. □
for Hermitian matrices A, B.
So we get the desired inequality. This completes the proof. □
for Hermitian matrices A, B.
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).
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