Integral Inequalities of Chebyshev Type for Continuous Fields of Hermitian Operators Involving Tracy–Singh Products and Weighted Pythagorean Means

In this paper, we establish several integral inequalities of Chebyshev type for bounded continuous fields of Hermitian operators concerning Tracy-Singh products and weighted Pythagorean means. The weighted Pythagorean means considered here are parametrization versions of three symmetric means: the arithmetic mean, the geometric mean, and the harmonic mean. Every continuous field considered here is parametrized by a locally compact Hausdorff space equipped with a finite Radon measure. Tracy-Singh product versions of the Chebyshev-Grüss inequality via oscillations are also obtained. Such integral inequalities reduce to discrete inequalities when the space is a finite space equipped with the counting measure. Moreover, our results include Chebyshev-type inequalities for tensor product of operators and Tracy-Singh/Kronecker products of matrices.


Introduction
One of the fundamental inequalities in mathematics is the Chebyshev inequality, named after P.L. Chebyshev, which states that for all real numbers a i , b i (1 i n) such that a 1 . . . a n and b 1 . . . b n , or a 1 . . . a n and b 1 . . . b n . This inequality can be generalized to where w i 0 for all 1 = 1, . . . , n. A matrix version of (2) involving the Hadamard product was obtained in [1]. A continuous version of the Chebyshev inequality [2] says that if f , g : If f and g are monotone in the opposite sense, the reverse inequality holds. In [3], Moslehian and Bakherad extended this inequality to Hilbert space operators related with the Hadamard product by using the notion of synchronous Hadamard property. They also presented integral Chebyshev inequalities respecting operator means.
The Grüss inequality, first introduced by G. Grüss in 1935 [4], is a complement of the Chebyshev inequality. This inequality gives a bound of the difference between the product of the integrals and the integral of the product for two integrable functions. For each integral function f : [a, b] → R, let us denote The Grüss inequality states that if f , g : [a, b] → R are integrable functions and there exist real constants k, K, l, L such that k f (x) K and l g(x) L for all x ∈ [a, b], then This inequality has been studied and generalized by several authors; see [5][6][7]. In [7], the term Chebyshev-Grüss inequalities is used mentioning to Grüss inequalities for Chebyshev functions T I which defined as A general form of Chebyshev-Grüss inequalities is given by where E is an expression depending on the arithmetic integral mean I and oscillations of f and g. Chebyshev-Grüss inequalities for some kind of operator via discrete oscillations is presented by Gonska, Raça and Rusu [7]. On the other hand, the notion of tensor product of operators is a key concept in functional analysis and its applications particularly in quantum mechanics. The theory of tensor product of operators has been investigate in the literature; see, e.g., [8,9]. In [10,11], the authors extend the notion of tensor product to the Tracy-Singh product for operators on a Hilbert space, and supply algebraic/order/analytic properties of this product.
In this paper, we establish a number of integral inequalities of Chebyshev type for continuous fields of Hermitian operators relating Tracy-singh products and weighted Pythagorean means. The Pythagorean means considered here are three classical means -the geometric mean, the arithmetic mean, and the harmonic mean. The continuous field considered here is parametrized by a locally compact Hausdorff space Ω endowed with a finite Radon measure. In Section 2, we give basic results on Tracy-Singh products for Hilbert space operators and Bochner integrability of continuous field of operators on a locally compact Housdorff space. In Section 3, we provide Chebyshev type inequalities involving Tracy-Singh products of operators under the assumption of synchronous Tracy-Singh property. In Section 4, we establish Chebyshev integral inequalities concerning operator means and Tracy-Singh products under the assumption of synchronous monotone property. Finally, we prove Chebyshev-Grüss inequalities via oscillations for continuous fields of operators in Section 5. In the case that Ω is a finite space with the counting measure, such integral inequalities reduce to discrete inequalities. Our results include Chebyshev-type inequalities concerning tensor product of operators and Tracy-Singh/Kronecker products of matrices.

Preliminaries
In this paper, we consider complex Hilbert spaces H and K. The symbol B(X) stands to the Banach space of bounded linear operators on a Hilbert space X. The cone of positive operators on X is denoted by B(X) + . For Hermitian operators A and B in B(X), the situation A ≥ B means that A − B ∈ B(X) + . Denote the set of all positive invertible operators on X by B(X) ++ .
We fix the following orthogonal decompositions: where all H i and K j are Hilbert spaces. Such decompositions lead to a unique representation for each operator A ∈ B(H) and B ∈ B(K) as a block-matrix form: where A ij ∈ B(H j , H i ) and B kl ∈ B(K l , K k ) for each i, j, k, l.

Tracy-Singh Product for Operators
Let A ∈ B(H) and B ∈ B(K). Recall that the tensor product of A and B, denoted by A ⊗ B, is a unique bounded linear operator on the tensor product space H ⊗ K such that When H = K = C, the tensor product of operators becomes the Kronecker product of matrices.
which is a bounded linear operator from When m = n = 1, the Tracy-Singh product A B is the tensor product A ⊗ B. If H i = K j = C for all i, j, the above definition becomes the usual Tracy-Singh product for complex matrices.  4. If A and B are Hermitian, then so is A B. 5. If A and B are positive and invertible, then (A B) α = A α B α for any α ∈ R. 6. If A C 0 and B D 0, then A B C D 0.

Bochner Integration
Let Ω be a locally compact Hausdorff (LCH) space equipped with a finite Radon measure µ. A family A = (A t ) t∈Ω of operators in B(H) is said to be bounded if there is a constant M > 0 for which A t M for all t ∈ Ω. The family A is said to be a continuous field if parametrization t → A t is norm-continuous on Ω. Every continuous field A = (A t ) t∈Ω can have the Bochner integral Ω A t dµ(t) if the norm function t → A t possess the Lebesgue integrability. In this case, the resulting integral is a unique element in for every bounded linear functional φ on B(H). Lemma 2 (e.g., [12]). Let (X, · X ) be a Banach space and (Γ, υ) a finite measure space. Then a measurable function f : Γ → X is Bochner integrable if and only if its norm function f is Lebesgue integrable. Lemma 3 (e.g., [12]). Let f : Γ → X be a Bochner integrable function. If ϕ : X → Y is a bounded linear operator, then the composition ϕ • f is Bochner integrable and , Proof. Since the map t → A t is continuous and bounded, it is Bochner integrable on Ω. Note that the map T → T X is linear and bounded by Lemma 1. Now, Lemma 3 implies that the map t → A t X is Bochner integrable on Ω and for all X ∈ B(K).

Chebyshev Type Inequalities Involving Tracy-Singh Products of Operators
From now on, let Ω be an LCH space equipped with a finite Radon measure µ. Let A = (A t ) t∈Ω , B = (B t ) t∈Ω , C = (C t ) t∈Ω and D = (D t ) t∈Ω be continuous fields of Hilbert space operators.

Definition 2.
The fields A and B are said to have the synchronous Tracy-Singh property if, for all s, t ∈ Ω, They are said to have the opposite-synchronous Tracy-Singh property if the reverse of (6) holds for all s, t ∈ Ω. 1. If A and B have the synchronous Tracy-Singh property, then 2. If A and B have the opposite-synchronous Tracy-Singh property, then the reverse of (7) holds.
Proof. By using Lemma 1, Proposition 1 and Fubini's Theorem [13], we have For the case 1, we have and thus (7) holds. For another case, we get the reverse of (8) and, thus, the reverse of (7) holds.

Remark 1.
In Theorem 1 and other results in this paper, we may assume that Ω is a compact Hausdorff space.
In this case, every continuous field on Ω is automatically bounded.
The next corollary is a discrete version of Theorem 1. 1. If A and B have the synchronous Tracy-Singh property, then 2. If A and B have the opposite-synchronous Tracy-Singh property, then the reverse of (9) holds.

Chebyshev Integral Inequalities Concerning Weighted Pythagorean Means of Operators
Throughout this section, the space Ω is equipped with a total ordering .

Definition 3.
We say that a field A is increasing (decreasing, resp.) whenever s t implies A s A t (A s A t , resp.).

Definition 4.
Two ordered pairs (X 1 , X 2 ) and (Y 1 , Y 2 ) of Hermitian operators are said to have the synchronous property if either X i Y i for i = 1, 2, or X i Y i for i = 1, 2.
The pairs (X 1 , X 2 ) and (Y 1 , Y 2 ) are said to have the opposite-synchronous property if either X 1 Y 1 and X 2 Y 2 , or X 1 Y 1 and X 2 Y 2 . Let us recall the notions of weighted classical Pythagorean means for operators. Indeed, they are generalizations of three famous symmetric operator means as follows. For any w ∈ [0, 1], the w-weighted arithmetic mean of A, B ∈ B(H) is defined by The w-weighted geometric mean and w-weighted harmonic mean of A, B ∈ B(H) ++ are defined respectively by For any A, B ∈ B(H) + , we define the w-weighted geometric mean and w-weighted harmonic mean of A and B to be respectively. Here, the limits are taken in the strong-operator topology. Lemma 4 (see e.g., [14]). The weighted geometric means, weighted arithmetic means and weighted harmonic means for operators are monotone in the sense that if A 1 A 2 and B 1 B 2 , then A 1 σB 1 A 2 σB 2 where σ is any of w , ! w , w .    Proof. Let s, t ∈ Ω and assume without loss of generally that s t. By applying Lemmas 1 and 5, Proposition 1, and Fubini's Theorem [13], we have If A, B, C, D are all increasing, we have by Lemma 4 that A t w C t A s w C s and B t w D t B s w D s . If A, B, C, D are all decreasing, we have A t w C t A s w C s and B t w D t B s w D s . Both cases lead to the same conclusion that and hence (10) holds. The cases 2.1 and 2.2 yield the same conclusion that and hence the reverse of (10) holds.
2. If (A, B) and (C, D) have the opposite-synchronous property, then the reverse of (11) holds.
Proof. For the synchronous case, we have by using positivity of the Tracy-Singh product (Lemma 1) that .
For the opposite-synchronous case, we have (A 1 − B 1 ) (A 2 − B 2 ) 0 and hence the reverse of inequality (11) holds.  Proof. Let s, t ∈ Ω and assume without loss of generally that s t. First, we consider the case 1. We have by using Lemmas 1 and 6, proposition 1, and Fubini's Theorem [13] that Now, by Lemmas 1 and 4, we have and hence (12) holds. The case 2 can be similarly proven.
Proof. Assume that (A, B) and (C, D) are synchronous. By continuity, we may assume that A, B, C, D > 0. We have Using Lemma 1 and (14), we get This implies that Hence, For the opposite-synchronous case, we have and hence the reverse of (13) holds.  Proof. Let s, t ∈ Ω with s t. If the pairs (A, B) and (C, D) are opposite-synchronous, then we have by applying Lemmas 1 and 7, Proposition 1, and Fubini's Theorem [13] that For the case 1, we have, by Lemmas 1 and 4, and hence (15) holds. Another assertion can be proved in a similar manner to that of the second assertion in Theorem 3.

Chebyshev-Grüss Inequaities via Oscillations
Throughout this section, let Ω be an LCH space equipped with a probability Radon measure µ.
We call osc(A) the oscillation of the field A.
Proof. We have by using Lemma 1, Proposition 1 and Fubini's Theorem [13] that Corollary 2. Let A i ∈ B(H) and B i ∈ B(K) be Hermitian operators for all i = 1, . . . , n. Then Proof. Set Ω = {1, . . . , n} equipped with the counting measure. We have We have

Conclusions
We establish several integral inequalities of Chebyshev type for continuous fields of Hermitian operators which are parametrized by an LCH space equipped with a finite Radon measure. We also obtain the Chebyshev-Grüss integral inequality via oscillations with respect to a probability Radon measure. These inequalities involve Tracy-Singh products and weighted versions of famous symmetric means. For a particular case that the LCH space is a finite space equipped with the counting measure, such integral inequalities reduce to discrete inequalities. Our results include Chebyshev-type inequalities for tensor product of operators and Tracy-Singh/Kronecker products of matrices.
Author Contributions: All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Funding: The first author expresses his gratitude towards Thailand Research Fund for providing the Royal Golden Jubilee Ph.D. Scholarship, grant no. PHD60K0225 to support his Ph.D. study.