We will discuss some operator inequalities on chaotic order about several operators, which are generalization of Furuta inequality and show monotonicity of related Furuta type operator function.

1. Introduction

An operator T is said to be positive (denoted by T ≥ 0) if (Tx, x) ≥ 0 for all vectors x in a Hilbert space, and T is said to be strictly positive (denoted by T > 0) if T is positive and invertible.

Theorem LH [Löwner-Heinz inequality, denoted by (LH) briefly] If A ≥ B ≥ 0 holds, then A^α ≥ B^α for any α ∈ [0,1].

This was originally proved in [1, 2] and then in [3]. Although (LH) asserts that A ≥ B ≥ 0 ensures A^α ≥ B^α for any α ∈ [0,1], unfortunately A^α ≥ B^α does not always hold for α > 1. The following result has been obtained from this point of view.

Theorem F (Furuta inequality). If A ≥ B ≥ 0, then for each r ≥ 0,

(i)
(B^r/2A^pB^r/2) ^1/q ≥ (B^r/2B^pB^r/2) ^1/q,
(ii)
(A^r/2A^pA^r/2) ^1/q ≥ (A^r/2B^pA^r/2) ^1/q

hold for p ≥ 0 and q ≥ 1 with (1 + r)q ≥ p + r.

The original proof of Theorem F is shown in [4], an elementary one-page proof is in [5], and alternative ones are in [6, 7]. We remark that the domain of the parameters p, q, and r in Theorem F is the best possible for the inequalities (i) and (ii) under the assumption A ≥ B ≥ 0; see [8].

We write A ≫ B if log A ≥ log B for A, B > 0, which is called the chaotic order.

Theorem A. For A, B > 0, the following (i) and (ii) hold:

(i)
A ≫ B holds if and only if A^r ≥ (A^r/2B^pA^r/2) ^r/(p+r) for p, r ≥ 0;
(ii)
A ≫ B holds if and only if for any fixed δ ≥ 0, F_A,B(p, r) = A^−r/2(A^r/2B^pA^r/2) ^{(δ + r)/(p+r)}A^−r/2 is a decreasing function of p ≥ δ and r ≥ 0.

(i) in Theorem A is shown in [9, 10], an excellent proof in [11], a proof in the case p = r in [12], (ii) in [9, 10], and so forth.

Lemma B (see [11].)Let A be a positive invertible operator, and let B be an invertible operator. For any real number λ,

()

Definition 1. Let A_n, A_n−1, …, A₂, A₁, B ≥ 0, r₁, r₂, …, r_n ≥ 0, and p₁, p₂, …, p_n ≥ 0 for a natural number n.

Let be defined by

()

For example,

()

Let q[n] be defined by

()

For example,

()

For the sake of convenience, we define

()

and these definitions in (6) may be reasonable by (2) and (4).

Lemma 2. For A_n, A_n−1, …, A₂, A₁, B ≥ 0 and any natural number n, we have

(i)
,
(ii)
q[n] = q[n − 1]p_n + r_n.

Proof. (i) and (ii) can be easily obtained by definitions (2) and (4).

2. Basic Results Associated with and q[n]

We will give some operator inequalities on chaotic order, and Theorem 5 is further extension of Theorem 3.1 in [13].

Lemma 3. If A ≫ B, for p ≥ 0 and r ≥ 0, then A ≫ (A^r/2B^pA^r/2) ^1/(p+r).

Proof. Since A ≫ B, we can obtain the following inequality.

A^r ≥ (A^r/2B^pA^r/2) ^r/(p+r) holds for p ≥ 0 and r ≥ 0 by (i) of Theorem A.

Take the logarithm on both sides of the previous inequality; that is,

()

therefor we have

()

Theorem 4. If A_n ≫ A_n−1 ≫ ⋯≫A₂ ≫ A₁ ≫ B and r₁, r₂, …, r_n ≥ 0, p₁, p₂, …, p_n ≥ 0 for a natural number n. Then the following inequality holds:

()

where

and q[n] are defined in (2) and (4).

Proof. We will show (9) by mathematical induction. In the case n = 1.

Since A₁ ≫ B implies

()

holds for any p₁ ≥ 0 and r₁ ≥ 0 by Lemma 3, whence (9) for n = 1.

Assume that (9) holds for a natural number k (1 ≤ k < n). We will show that (9) holds r₁, r₂, …, r_k, r_k+1 ≥ 0 and p₁, p₂, …, p_k, p_k+1 ≥ 0 for k + 1.

Put D = A_k+1, E = A_k, and , and (9) holds for n = k implying

()

Equation (11) yields the following by Lemma 3, for r ≥ 0 and p ≥ 0:

()

that is,

()

Put r = r_k+1, p = q[k]p_k+1 in (13), then by (ii) of Lemma 2, the exponential power 1/(p + r) of the right hand side of (13) can be written as follows:

()

and we have the following desired (15) by (12) and (13):

()

so that (15) shows that (9) holds for k + 1.

Theorem 5. If A_n ≫ A_n−1 ≫ ⋯≫A₂ ≫ A₁ ≫ B and r₁, r₂, …, r_n ≥ 0 for a natural number n. For any fixed δ ≥ 0, let p₁, p₂, …, p_n be satisfied by

()

The operator function I_k(p_k, r_k) for any natural number k such that 1 ≤ k ≤ n is defined by

()

Then the following inequality holds:

()

for every natural number k such that 1 ≤ k ≤ n, where

and q[n] are defined in (2) and (4).

Proof. Since , q[0] = 1 in (6), we may define I₀(p₀, r₀) = B^δ for p₀ = r₀ = 0.

Because A₁ ≫ B, then for any fixed δ ≥ 0,

()

since

holds by (ii) of Theorem A. And (19) can be expressed as

()

We can apply Theorem 4, and we have the following (21) for any natural number k such that 1 ≤ k ≤ n:

()

Since X ≫ Y implies that X^t ≫ Y^t holds for any t ≥ 0, (21) ensures

()

Putting

and applying (19) for δ = 1 and A ≫ B₁, we have

()

holds for p ≥ 1 and r ≥ 0.

Putting r_k+1 = r(δ + r₁ + r₂ + ⋯+r_k) in (23), then (23) can be rewritten by

()

Putting p = (q[k]p_k+1)/(δ + r₁ + r₂ + ⋯+r_k) ≥ 1, since p_k+1 ≥ (δ + r₁ + r₂ + ⋯+r_k)/q[k] in (16), then we have

()

and we have (18) for k such that 1 ≤ k ≤ n by (25) and (20) since (20) means (18) for k = 1.

Corollary 6. If A_n ≫ A_n−1 ≫ ⋯≫A₂ ≫ A₁ ≫ B and r₁, r₂, …, r_n ≥ 0 for a natural number n. For any fixed δ ≥ 0, let p₁, p₂, …, p_n be satisfied by (16).

Then the following inequalities hold:

()

where

, q[n], and I_k(p_k, r_k) (1 ≤ k ≤ n) are defined in (2), (4), and (17).

Proof. Applying (18) of Theorem 5 for k such that 1 ≤ k ≤ n, we have

()

3. Monotonicity Property on Operator Functions

We would like to emphasize that the condition of Theorem 7 is stronger than Theorem 5, and moreover when we discuss monotonicity property on operator functions, we can only apply Theorem 7.

Theorem 7. If A_n ≫ A_n−1 ≫ ⋯≫A₂ ≫ A₁ ≫ B and r₁, r₂, …, r_n ≥ 0, p₁, p₂, …, p_n ≥ 0 for a natural number n. Then the following inequality holds:

()

where

and q[n] are defined in (2) and (4).

Proof. We will show (28) by mathematical induction. In the case n = 1.

Since A₁ ≫ B implies

()

holds for any, p₁ ≥ 0 and r₁ ≥ 0 by (i) of Theorem A, whence (28) for n = 1.

Assume that (28) holds for a natural number k (1 ≤ k < n). We will show (28) for r₁, r₂, …, r_k+1 ≥ 0 and p₁, p₂, …, p_k, p_k+1 ≥ 0 for k + 1.

We can obtain the following inequality from the hypothesis (28) for the case n = k:

()

hence we have

, and (i) of Theorem A ensures

()

Putting r = r_k+1 and p = q[k]p_k+1, then we have the following inequality:

()

so that (32) shows (28) for k + 1.

Theorem 8. If A_n ≫ A_n−1 ≫ ⋯≫A₂ ≫ A₁ ≫ B and r₁, r₂, …, r_n ≥ 0 for a natural number n. For any fixed δ ≥ 0, let p₁, p₂, …, p_n be satisfied by (16).

Then

()

is a decreasing function of both r_n ≥ 0 and p_n which satisfies

()

where

and q[n] are defined in (2) and (4).

Proof. Since the condition (16) with δ ≥ 0 suffices (28) in Theorem 7, we have the following inequality by Theorem 7; see (28).

We state the following important inequality (35) for the forthcoming discussion which is the inequality in (16):

()

because the inequality in (35) follows by (ii) of Lemma 2, and the inequality follows by

()

obtained by (34).

(a) Proof of the result that I_n(p_n, r_n) is a decreasing function of p_n.

Without loss of generality, we can assume that p_n > 0. We can obtain the following inequality by (28) and by (i) of Lemma 2:

()

and (37) implies

()

Put α = ω/p_n ∈ [0,1] for p_n ≥ ω ≥ 0, then we raise each side of (38) to the power α = ω/p_n ∈ [0,1], then

()

Whence we have

()

and the last inequality holds by LH because (39) and (δ + r₁ + r₂ + ⋯+r_n)/(q[n − 1](p_n + ω) + r_n) ∈ [0,1] which is ensured by (35) and q[n] + q[n − 1]ω = q[n − 1](p_n + ω) + r_n ≥ q[n] by (4), so that I_n(p_n, r_n) is a decreasing function of p_n.

(b) Proof of the result that I_n(p_n, r_n) is a decreasing function of r_n.

Without loss of generality, we can assume that r_n > 0. Raise each side of (28) to the power μ/r_n ∈ [0,1] for r_n ≥ μ ≥ 0 by LH, then

()

We state the following inequality by (ii) of Lemma 3 and (35):

()

Then we have

()

and the last inequality holds by LH because (41) and

()

so that I_k(p_k, r_k) is a decreasing function of r_n.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (1127112; 11201127), Technology and Pioneering project in Henan Province (122300410110).

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All articles

Some Operator Inequalities on Chaotic Order and Monotonicity of Related Operator Function

Abstract

1. Introduction

2. Basic Results Associated with and q[n]

3. Monotonicity Property on Operator Functions

Acknowledgments

References

References

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