On the range of the parameters for the grand Furuta inequality to be valid
© Watanabe; licensee Springer. 2014
Received: 3 March 2014
Accepted: 8 July 2014
Published: 19 August 2014
In order to investigate the precise range D of the parameters p, s, t, and r for which the grand Furuta inequality is valid. We use the method of reductio ad absurdum. We find the following results: The area , , , and is contained in the complement of the range D. The condition seems essential for the grand Furuta inequality.
KeywordsLöwner-Heinz inequality Furuta inequality matrix inequality
A bounded linear operator T on a Hilbert space is said to be positive semidefinite (denoted by ) if for all vectors h. We write if T is positive semidefinite and invertible.
Theorem 1.1 
It is well known that Theorem 1.1 is equivalent to the next theorem, which is often called the essential case of the Furuta inequality.
as far as all parameters are positive. We would like to emphasize that the theorem can be divided into two cases.
Theorem 1.3 
The next proposition is corresponding to the case of the previous theorem by putting .
On the other hand, the following ‘α-free’ proposition corresponds to the case of Theorem 1.3 by putting .
Since the condition is the essential case for the Furuta inequality, our interest in Proposition 1.5 is not at all less than Proposition 1.4.
Furuta gave a unifying extension of both Theorem 1.1 and the Ando-Hiai inequality , which is often called the grand Furuta inequality.
Theorem 1.6 
Again, Tanahashi showed that the outside powers in this theorem are the best possible.
Theorem 1.7 
Remark 1.8 In , Theorem 1.7 is originally stated as follows:
Although it would be a nice theorem if one could precisely determine the range (2) all at once, it seems difficult to the author. Therefore, we should treat several main cases of the problem.
It is quite natural to expect an ‘α-free’ version which can be regarded as corresponding to Proposition 1.5. The following result obtained by Koizumi and the author is such an attempt.
Theorem 1.9 
Then there exist matrices A, B with that do not satisfy the inequality (1).
If , , , and , then .
If , , , and , then .
Remark 1.11 The case (ii) of [, Theorem 2.1] by Koizumi and the author treats the case , , . However, we have by the notations in  and the proof for (i) is not applicable to (ii). It seems still open.
The main purpose of this article is to show that the area , , , and is contained in the complement of the range (2).
where k is a positive scalar to be specified later.
We begin with an elementary inequality for real numbers.
As Theorem 1.7 is regarded as an extension of Proposition 1.4, our main theorem can be considered as an extension of Proposition 1.5, which shows one of the largest pieces of the complement of the range (2). The advantage is that the assumptions on the parameters other than are very mild. Note that, if we change to , then the inequality holds for .
Yamazaki’s simplified proof of Theorem 1.7 in  is not applicable in our context. The method of our proof is the same as Tanahashi’s argument, whose outline is explained in the previous section. We would like to emphasize there are several branching points such that the conditions about parameters in the assumption are to be reflected to powers or coefficients in calculations. Moreover, we have to estimate all terms up to the order of in the sequel. On the other hand, in the existing literature, such as [7, 8] and , it is sufficient to estimate only main and second terms.
by the notation of the previous section. It is easy to see that , , , and as . Therefore, for sufficiently large x, we have . Further, we shall see later , as . Since , we have for sufficiently large x.
where k is the nonnegative integer determined by .
This is a contradiction and completes the proof of Theorem 3.2. □
Proof It is obvious that and . □
The author was supported in part by Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science.
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