- Open Access
Unitarily invariant norm inequalities for some means
© Furuichi; licensee Springer. 2014
- Received: 8 January 2014
- Accepted: 4 April 2014
- Published: 6 May 2014
We introduce some symmetric homogeneous means, and then we show unitarily invariant norm inequalities for them, applying the method established by Hiai and Kosaki. Our new inequalities give tighter bounds of the logarithmic mean than the inequalities given by Hiai and Kosaki. Some properties and norm continuities in the parameter for our means are also discussed.
- symmetric homogeneous mean
- logarithmic mean
- unitarily invariant norm and norm inequality
In the previous paper, we derived tight bounds for the logarithmic mean in the case of the Frobenius norm, inspired by the work of Zou in .
Theorem 1.1 ()
In this paper, we give tighter bounds for the logarithmic mean than those by Hiai and Kosaki [3, 4] for every unitarily invariant norm. That is, we give the generalized results of Theorem 1.1 for the unitarily invariant norm. For this purpose, we firstly introduce two quantities.
We here define a few symmetric homogeneous means using and in the following way.
- (i)For and , we define
- (ii)For and , we define
which is independent of α.
- (iii)For and , we define
- (iv)For and , we define
and we also set for .
and are called Stolarsky mean and binomial mean, respectively.
which equivalently implied Frobenius norm inequalities (Theorem 1.1). See Theorem 2.2 and Theorem 3.2 in  for details. In this paper, we give unitarily invariant norm inequalities which are general results including Frobenius norm inequalities as a special case.
is non-decreasing in x, y.
The functions , , , defined in Definition 1.4 are symmetric homogeneous means. We give powerful theorem to obtain unitarily invariant norm inequalities. In the references [4–7], another equivalent conditions were given. However, here we give minimum conditions to obtain our results in this paper. Throughout this paper, we use the symbol as the set of all bounded linear operators on a separable Hilbert space ℋ. We also use the notation if satisfies for all (then K is called a positive operator).
for any with and for any unitarily invariant norm .
The function is positive definite function on ℝ (then we denote ), where the positive definiteness of a real continuous function ϕ on ℝ means that is positive definite for any with any .
Thanks to Theorem 2.2, our task to obtain unitarily invariant norm inequalities in this paper is to show the relation , which is stronger than the usual scalar inequalities . That is, implies .
We firstly give monotonicity of three means , and for the parameter . Since we have , and , we consider the case . Then we have the following proposition.
If , then .
If , then .
If , then .
- (i)We calculate
This is a positive definite function for the case , so that we have .
- (ii)The similar calculation
- (iii)Since the case follows from the limit of the case , we may assume . Since we have
From Proposition 4 in , the sufficient condition that the function is positive definite is , i.e., . The sufficient condition that the function is positive definite is . The conditions and are satisfied with a natural number n sufficiently large. Thus we conclude .
It may be notable that (iii) of the above proposition can be proven by a similar argument to Theorem 2.1 of the paper .
Next we give the relation among the four means , , , and .
which is a positive definite function. Thus we have so that the first inequality of this proposition holds thanks to Theorem 2.2.
implies . Thus we have the second inequality of this proposition.
implies . Thus we have the third inequality of this proposition. □
(More concrete expressions of these inequalities will be written down in (8) of Section 4.)
Proof The second inequality and the third inequality have already been proven in Proposition 2.4.
By Proposition 5 in , the function is positive definite, if and . The function is also positive definite, if . The case and satisfies the above conditions. Thus we have which leads to the first inequality of this proposition.
By Proposition 5 in , the function is positive definite, if and . The function is also positive definite, if . From these conditions, we have and . The case and satisfies the above conditions. Thus we have , which leads to the last inequality. □
which means that Theorem 2.5 gives a general result for Theorem 1.1.
for any with , and any unitarily invariant norm .
However, we do not have the scalar inequality for in general, so that the inequality (2) is not true for . We also do not have the scalar inequality for , in general.
In this section, we consider the norm continuity argument with respect to the parameter on our introduced means. Since we have the relation , we firstly consider the norm continuity in the parameter on .
by the Lebesgue dominated convergence theorem. □
We secondly consider the norm continuity in the parameter on .
which is the second inequality of (3).
Thus we have the inequality (4). □
We also have the following proposition.
which implies (7) for the case .
To prove , we have only to prove thanks to Lemma 5.8 in . Since we have , we have in the limit . From the fact for , we also have . Since and are positive definite functions, we have and . (See Chapter 5 in  for some basic properties of the positive definite function.) We thus obtain for two -functions and . We finally obtain . Since is integrable and , we obtain by the Lebesgue dominated convergence theorem. □
We note that the assumption for some is equivalent to , since we have using the inequality (3).
for with , , , and any unitarily invariant norm .
We have also shown some properties for our means such as monotonicities and norm continuities in the parameter.
I would like to express my deepest gratitude to Professor Fumio Hiai and Professor Hideki Kosaki for giving me valuable comments to improve this manuscript. I also was partially supported by JSPS KAKENHI Grant Number 24540146.
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