Generalization of the Lieb–Thirring–Araki Inequality and Its Applications

: The matrix eigenvalue is very important in matrix analysis, and it has been applied to matrix trace inequalities, such as the Lieb–Thirring–Araki theorem and Thompson–Golden theorem. In this manuscript, we obtain a matrix eigenvalue inequality by using the Stein–Hirschman operator interpolation inequality; then, according to the properties of exterior algebra and the Schur-convex function, we provide a new proof for the generalization of the Lieb–Thirring–Araki theorem and Furuta theorem.


Introduction
As an important branch of mathematics, matrix theory has been widely applied in the fields of mathematics and technology, such as optimization theory ( [1]), differential equations ( [2]), numerical analysis, operations ( [3]) and quantum theory ( [4]).
In this manuscript, let C n be an n-dimensional complex vector space with the inner product x, y = x * y = ∑ n i=1 x * i y i for x = (x 1 , · · · , x n ) , y = (y 1 , · · · , y n ) ∈ C n , where the superscripts x * and denote the conjugated transpose of x and the matrix transpose, respectively. Let M n denote the whole set of n × n matrices with complex entries, and we call x ∈ C n the eigenvector of A ∈ M n when A x = λ x (where λ is called the eigenvalue of A). We denote H n the set of all Hermitian matrices. For any A ∈ H n , we have A = ∑ n i=1 λ i P i , where λ i is the eigenvalue of A and ∑ n i=1 P i = Id, P i P j = 0(i = j); specially, when x * A x ≥ 0 for any x ∈ C n , we denote A ∈ H + n (H + n is the set of n × n positive-definite Hermitian matrices whose eigenvalues are nonnegative). Let f be a function with the domain (0, +∞); for any A ∈ H + n , the matrix function is defined as On the basis of this definition, we have a formula relating the trace of matrix A and the eigenvalue of A: where λ i (A) is the eigenvalue of A. It is well known that Tr [ Thompson and Golden independently discovered an inequality called the Thompson-Golden theorem (refer to [5][6][7]): In general, the following limit holds (called the Lie-Trotter formula [8] Furthermore, the following inequality holds when p ≥ 1: which is the Lieb-Thirring-Araki theorem ( [9,10]). Since the function F(A) = Tr e B+ln A is a Fréchet differential function for any A ∈ H n , the concavity of F(A) implies the Thompson-Golden theorem. At the same time, one can also obtain the Thompson-Golden theorem by using the relationship p . By using the matrix exterior algebra, we have According to the convexity of Tr ∧ k e A , Huang proved the following inequality ( [12]): With this motivation, we utilize the Stein-Hirschman operator interpolation inequality to show that λ 1 1 α is a monotone increasing function for any α > 0. Then, we generalize the Lieb-Thirring-Araki theorem and provide a new proof of the Furuta theorem ([13]). The rest of the paper is organized as follows. In Section 2, some general definitions and important conclusions are introduced. In Section 3, a new proof of the monotonicity of λ 1 1 α and some general results are offered.

Preliminary
In this section, we recall some notions and definitions from matrix analysis, and introduce some important results of the matrix-monotone function, which are used through the article (refer to [14][15][16][17]).

Tensor Product and Exterior Algebra
The tensor product, denoted by ⊗, is also called the Kronecker product. It is a generalization of the outer product from vectors to matrices, so the tensor product of matrices is referred to as the outer product as well in some contexts. For an m × n matrix A and a p × q matrix B, the tensor product of A and B is defined by where A = a ij 1≤i≤m,1≤j≤n . The tensor product is different from matrix multiplication, and one of the differences is commutativity: From this relation, one can obtain For convenience, we denote In addition to the tensor product, there is another common product named exterior algebra ( [18]). Exterior algebra, denoted by ∧, is a binary operation for any A n×n that is where {ξ j } n j=1 is an orthogonal basis of C n and where σ n is the family of all permutations on {1, 2, · · · , n}. Let k C n be the span of the {ξ i 1 ∧ ξ i 2 · · · ∧ ξ i k } 1≤i 1 <···<i k ≤n ; a simple calculation shows that (1)

Schur-Convex Function
Let x = (x 1 , · · · , x n ) , y = (y 1 , · · · , y n ) ∈ R n and denote then x is said to be majorized by y, denoted by Suppose f is a real-valued function defined on a set A ⊆ R n ; then, f is said to be a Schur-convex function on A if, for any x, y ∈ R n and x ≺ y, one obtains f ( x) ≤ f ( y) ( [11]).
If f is differentiable and defined on I n (I ⊂ R being an open interval), then the following lemma holds (refer to [11]).

The Matrix-Monotone Function
For a matrix A ∈ H + n , according to the spectral theorem ( [19]), it can be decomposed as where P is the unitary matrix and Λ A := diag{λ 1 , ..., λ n } is a diagonal matrix with eigenvalues as elements. When Associated with a function f (x) on (0, +∞), the matrix function f (A) is defined as Since the matrix-monotone function is a special type of operator monotone function, we present the following general conclusion about the operator-monotone function, which can be found in [20,21].

Lemma 2.
The following statements for a real-valued continuous function f on (0, +∞) are equivalent: ( i) f is operator-monotone; ( ii) f admits an integral representation where α is a real number, β is non-negative and µ is a finite positive measure on (−∞, 0).

The Main Results
For any A, B ∈ M n , it is known ln AB is not equal to ln A + ln B in general when Therefore, many people pay much attention to studying the relation between (A A famous result regarding the trace inequality is the Lieb-Thirring-Araki theorem: where α ≥ 1. In the following, we further study the relation between (A Theorem 1. For any 0 < α ≤ β and A, B ∈ H + n , the following inequality holds Proof. By using the Cauchy inequality, we have If we denote as the maximum eigenvalue of A, then we can obtain Here, we use the fact that λ 1 (AB) = λ 1 (BA). Through a simple deformation, we have k . From this inequality, we have the expression This implies Namely, for 0 < α ≤ β, we obtain This completes the proof of Theorem 1.
Although Theorem 1 has been obtained from the Cauchy inequality, the frequency of retractions improves the inequality. In the following, we obtain Theorem 1 by using operator interpolation. First, let us introduce the Stein-Hirschman operator interpolation inequality ( [12]).
From Lemma 3, we can improve the result in Theorem 1 and obtain the following theorem.

Theorem 2.
For any A, B ∈ H + n , the following inequality holds Proof. Firstly, let f be an analytic function in C.
then, for any f ∈ L 1 t (C), we can obtain This implies for any 0 < t < 1, and the first "≤" is obtained by the Jensen inequality ( [22]). This completes the proof of Theorem 2.
Theorem 2 is very useful. On one hand, when α < β, letting t = α β , we can obtain Theorem 1. On the other hand, using the matrix exterior algebra, we obtain Furthermore, we can deduce the following inequality whether it is true or not for any k ≤ n, and this inequality can be regarded as a generalization of the Lieb-Thirring-Araki theorem.

Generalization of Lieb-Thirring-Araki Theorem
According to Theorem 1 and Formula (1), we can show that 1 β ], (6) when α ≤ β. Specially, when β = 1 and We know that the Lieb-Thirring-Araki theorem can be obtained from the Schur-convex Generally, we can prove the following conclusion.
From Theorem 3, we can deduce the following inequality immediately.

Corollary 1.
For any α ≥ 1 and A, B > 0, the following inequality holds or for any γ > 0.
From (8), it can be seen that, when k = 1, Corollary 1 is just the Lieb-Thirring-Araki theorem. Especially, Tr ∧ k (A Using Theorem 1, we can obtain for any 0 ≤ α ≤ β, and this is a generalization of the Thompson-Golden theorem. For some other generalizations of the Thompson-Golden theorem, see [8,23]. Moreover, since where 0 < α ≤ 1 and r ≥ 1, we can obtain the following corollary.

Corollary 2.
For any r ≥ 1 and A, B > 0, the following inequality holds:

Applications in Matrix-Monotone Function
In this subsection, we obtain some other corollaries from Theorem 1 associated with the matrix-monotone function. Since Hence, we obtain the Löwner-Heinz Theorem ( [4]).

This implies
That is, 3 2 ]. Repeating this process, we have finished the proof.

Some Other Applications
In this subsection, we obtain a corollary associated with the matrix determinant. We suppose A, B ∈ H n and λ n (e where 0 < α ≤ β and ln λ i (e A 2 e B e A 2 ) ≥ 0 (i = 1, 2 · · · , n). Let Then, a straightforward calculation indicates Hence, d(x 1 , · · · , x n ) is a Schur-concave function and the following inequality holds ( [8]).
In fact, for any A ∈ H n , we have Hence, Corollary 5 can be generalized as the following corollary.

Proof.
Since we can finish the proof if we show that the function a(x 1 , x 2 · · · , x n ) = ∑ 1≤i 1 <i 2 <···<i k ≤n x i 1 x i 2 · · · x i k is Schur-concave for any x i ≥ 0. In fact, we have This completes the proof of Corollary 6.

Conclusions
In the paper, we discuss the relationship between λ 1 (A 1 2 BA 1 2 ) α and λ 1 (A α 2 B α A α 2 ) by using the Stein-Hirschman operator interpolation inequality. Through in-depth study, we obtain some eigenvalue inequalities such as the generalization Golden-Thompson theorem and Lieb-Thirring-Araki theorem. Moreover, the Furuta theorem is also shown by using the eigenvalue inequality. At last, we generalize an important determinant inequality by using the matrix exterior algebra.