#### Abstract

This work is to consider Furuta type inequalities and their applications. Firstly, some Furuta type inequalities under are obtained via Loewner-Heinz inequality; as an application, a proof of Furuta inequality is given without using the invertibility of operators. Secondly, we show a unified satellite theorem of grand Furuta inequality which is an extension of the results by Fujii et al. At the end, a kind of Riccati type operator equation is discussed via Furuta type inequalities.

#### 1. Introduction

Throughout this paper, an operator means a bounded linear operator on a Hilbert space. and mean a positive operator and an invertible positive operator, respectively, (see [1, page 103]). The classical Loewner-Heinz inequality (L-H) is stated below (see [2, page 127]).

Theorem 1 (Loewner-Heinz inequality (L-H)). *Let ; then ensures
*

In general, (L-H) is not true for . As a celebrated development of (L-H), Furuta provided a kind of order preserving operator inequality [2, page 129], the so-called Furuta inequality (FI).

Theorem 2 (Furuta inequality (FI), [3]). *Let , ; then ensures
*

Tanahashi proved that the outer exponent above is optimal; see [3] for related topics. In order to establish the order structure on Aluthge transform of nonnormal operators, the complete form of Furuta inequality was showed in [4].

Theorem 3 (Complete form [4]). *Let , , and . Then and such that ensures
*

We call the theorem above the complete form of Furuta inequality because the case of it implies the essential part () of Furuta inequality by the Loewner-Heinz inequality for . For convenience, we call Furuta inequality (Theorem 2) the original form of Furuta inequality.

It is known that there are many applications of Furuta type inequalities; we cite [5–7].

Based on Ito et al. [8] which is a continuation of [9], the equivalent relations between two operator inequalities are useful. For , means the projection .

Theorem 4 (see [8]). *Let , , and .*(1)*If , then, for each , , and , the following inequalities are equivalent to each other:
* *In particular, (4) implies (5) without condition .*(2)*For each , , and , the following inequalities are equivalent to each other:
*

It should be pointed out that (5) ensures (4) is not true without the condition [8, Remark 1]. Moreover, the proof of Theorem 4 is independent of (L-H).

In Section 2, some Furuta type inequalities under are proved via Loewner-Heinz inequality; as applications, we show alternate proofs of some well-known Furuta type inequalities (proofs of Theorems 10 and 2).

In 1995, Furuta [10] proved the so-called grand Furuta inequality which is also an extension of Theorem 2.

Theorem 5 (grand Furuta inequality [10]). *Let , , and . If with ; then
*

Fujii et al. proved some satellite theorems of grand Furuta inequality.

Theorem 6 (see [11]). *Let , , and . If with ; then
*

Theorem 7 (see [12]). *Let , , and . If with ; then
*

Theorems 6 and 7 are extensions of Theorem 5.

In Section 3, we will show a unified satellite theorem which is an extension of Theorems 6 and 7 via the complete forms of Furuta inequality with negative powers.

Lastly, it is known that Riccati type operator equations relate to control theory closely and have been studied extensively [13]. Pedersen and Takesaki [14] developed the special kind of Riccati equation as a useful tool for the noncommutative Radon-Nikodym theorem.

Yuan and Gao [15] discussed the Riccati type equation:

In Section 4, as a continuation of [15, 16], we will consider the Riccati type equation: via Furuta type inequalities.

#### 2. Furuta Type Inequalities under the Order

Reference [17] proved a kind of equivalent relations which can be regarded as a parallel result to Theorem 4.

Theorem 8 (see [17]). *Let , , and . If , then, for each , and , the following inequalities are equivalent to each other:
**
In particular, (12) implies (13) without condition .*

The proof of Theorem 8 is different from Theorem 4 and independent of (L-H).

In this section, we consider some Furuta type inequalities under the order . As applications, alternate proofs of some Furuta type inequalities are given (proofs of Theorems 10 and 2). Especially, we prove (FI) without using the invertibility of operators.

Theorem 9. *Let , . *(1)*For each and with , the following inequalities hold and they are equivalent to each other:
*(2)*For each and with , the following inequalities hold:
*(3)*If , then
*

*Proof. *(1) Since , follows. By and (L-H) for and , we have
Hence, (14) holds. Since , follows. So, the equivalency follows by Theorem 8.

(2) Similar to the proof of (14), we have
Hence, (16) holds. Since (12) implies (13) without kernel condition, (17) follows by (16).

(3) By (15), there exists the function defined on satisfying [18, Lemma 2.6(1)]. Hence, case of [18, Lemma 2.6(2)] implies
So (18) holds by and (L-H) for . It is easy to prove (19) in a similar way.

As prompt applications, we show alternate proofs of some Furuta type inequalities.

Theorem 10 (see [19]). *Let , , , . For such that , if
**
then
**
where . Moreover, for each , the function
**
is decreasing (resp., increasing) for .*

*Proof. *It is enough to prove the case because the case can be proved in a similar manner. Denote (23) by ; that is,
For , , by (19) of Theorem 9, we have
By putting , the inequality above becomes
This implies that (24) holds for . Denote and ; repeating this process, (24) holds for .

For each , , by (24) and (L-H),
where . This together with Theorem 8 and (L-H) deduce that
where . So, the monotonicity of the function holds.

It should be pointed out that, if and , the assertion that (23) ensures (24) is not true [15, Theorem 2.8].

Theorem 11 (see [15]). *Given any positive numbers , , , and with , there exist invertible positive operators and such that
**
where is an arbitrary positive number.*

*Alternate Proof of Theorem 2*. The case and of Theorem 2 follows by (L-H) directly. Theorem 9(3) means the case and of Theorem 2; this together with Theorem 10 implies the case and of Theorem 2. So, the proof is complete.

The proof above says that the original form of Furuta inequality (Theorem 2) is a composition of (L-H), Theorems 9 and 10. The proof here is independent of the invertibility of the operators and .

#### 3. A Unified Satellite Theorem of Grand Furuta Inequalities

Denote , where .

Theorem 12. *Let , , , with .*(1)*If , , and , then
*(2)*If and , then
*

The case of Theorem 12(2) is just Theorem 7. The special case of Theorem 12(1) implies the result below.

Corollary 13. *Let , , , with . If and , then
*

It is obvious that the special case of Corollary 13 is a unified result of Theorems 6 and 7; that is, it is an extension of Theorems 6 and 7. So, we call Theorem 12 a unified satellite theorem of grand Furuta inequality (Theorem 5).

In order to give a proof, we prepare some results in advance.

Lemma 14 (see [18]). *Let , and . Then with ensures that the function
**
is decreasing for . In particular,
*

Lemma 15 (see [18]). *Let , , , and . Then with ensures
*

Lemma 16. *Let , and . Then the following assertion (1) implies (2).*(1)

*There exists an increasing function such that, for each , if , then*(2)

*The function in*

*(1)*satisfies that, for each , if , thenLemma 16 is a complement to [18, Lemma 2.6].

*Proof. *It is sufficient to prove the case for the case can be proved in a similar manner. For each and , if , then (2) follows by (1) immediately. Suppose that for some positive integer and . By , for , we have
Noting that and , these together with (L-H) deduce that
Therefore, the function in satisfies .

Lemma 17. *Let , and ; then with ensures
*

*Proof. *Firstly, we prove the case of Lemma 17. By [10, Lemma 1], (42) is equivalent to
On the other hand, holds by Loewner-Heinz inequality for . So (42) holds for .

Now, it is proved that (42) holds when . Meanwhile, it is easy to see that the increasing function satisfies of Lemma 16, so (42) holds for .

*Proof of Theorem 12. *By the case of Lemma 15 and (L-H) for ,
(1)For , Theorem 3 and (L-H) deduce that
Meanwhile, for and , Lemma 17 and (L-H) imply
Hence, follows by the case of (44), (45), and (46).(2)By (L-H), (44), , Theorem 3 and Lemma 17 ensure
The above is the same as the function in Lemma 14.

#### 4. Riccati Type Operator Equations

Yuan and Gao [15] discussed the Riccati type equation:

Theorem 18 (see [15]). *Let , and assume that .*(1)*The following statements are equivalent for each , and .(a) for some .(b)There exists a unique operator that satisfies and (48).*

*If in additional is invertible, (1) holds for .*(2)

*If there exists satisfying (48) for fixed , and , then, for and , there exists satisfying*

One of the applications of Riccati equation (48) is to show that the inclusion relations among class operators are strict [15, Theorem 3.1]. Recently, there are some developments on operator equations including the following equation (see [16, 20]):

Obviously, the special case of (50) is just (48).

Theorem 19 (see [16]). *Let , and assume that . The following statements are equivalent for each , , and .*(1)* for some .*(2)*There exists a unique operator which satisfies and (50).**
If in additional is invertible, the condition can be replaced with where means the set of all real numbers, and if and are both invertible, the conditions and can be replaced with and .*

The case of Theorem 19 is a generalization of Theorem 18(1). In this section, we give a generalization of Theorem 18(2).

Lemma 20. *Let , , , , , . For and such that and , if
**
then, for ,
**
where .*

The case , and of Lemma 20 implies Theorem 10. The case , and of Lemma 20 implies Yanagida's result [21, Proposition 4].

*Proof. *It is enough to prove the case because the case can be proved in a similar manner. Denote (51) by ; that is,
For , , by (FI) (Theorem 2), we have
By putting , the inequality above becomes
Denote and ; then and , so that (52) holds by the inequality above.

Theorem 21. *Let , and assume that . For each , and , if there exists satisfying the equation
**
where , and . Then, for and , there exists satisfying
**
If is invertible, the condition can be replaced with .*

*Proof. *By the assumption, (1) of Theorem 19 holds for some ; that is,
So, the following holds by Lemma 20:
where ; that is,
where . Hence, (57) is solvable by Theorem 19.

The result below is the case and of Theorem 21.

Corollary 22. *Let , and assume that . For each , and , if there exists satisfying the equation
**
where and . Then, for , there exists satisfying
**
If is invertible, the condition can be replaced with .*

It is obvious that Corollary 22 is a generalization of the case of Theorem 18(2).

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (11301155) and Project of Education Department of Henan Province of China (2012GGJS-061), and Project of Science and Technology Department of Henan Province of China (142300410143).