A converse of Loewner–Heinz inequality and applications to operator means

J. Math. Anal. Appl. 413 (2014) 422–429

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A converse of Loewner–Heinz inequality and applications to operator means ✩ Mitsuru Uchiyama a , Takeaki Yamazaki b,∗ a b

Department of Mathematics, Shimane University, Matsue City, Shimane, Japan Department of Electrical, Electronic and Computer Engineering, Toyo University, Kawagoe 350-8585, Japan

a r t i c l e

i n f o

a b s t r a c t Let f (t ) be an operator monotone function. Then A  B implies f ( A )  f ( B ), but the converse implication is not true. Let A  B be the geometric mean of A , B  0. If A  B, then B −1  A  I; the converse implication is not true either. We will show that if f (λ B + I )−1  f (λ A + I )  I for all sufficiently small λ > 0, then f (λ A + I )  f (λ B + I ) and A  B. Moreover, we extend it to multi-variable matrices means. © 2013 Elsevier Inc. All rights reserved.

Article history: Received 26 July 2013 Available online 3 December 2013 Submitted by J.A. Ball Keywords: Positive definite operators Loewner–Heinz inequality Operator mean Operator monotone function Operator concave function Ando–Hiai inequality Geometric mean Karcher mean Power mean

1. Introduction In what follows, H means a complex Hilbert space with inner product ·,·, and an operator means a bounded linear operator on H. An operator A is said to be positive (denoted by A  0) if and only if  Ax, x  0 for all x ∈ H, and A  B means B − A is positive. Moreover, an operator A is said to be positive definite (denoted by A > 0) if A is positive and invertible. A real continuous function f (t ) defined on a real interval I is said to be operator monotone, provided A  B implies f ( A )  f ( B ) for any two bounded self-adjoint operators A and B whose spectra are in I . The Loewner–Heinz inequality means the power function t a is operator monotone on [0, ∞) for 0 < a < 1. log t is operator monotone on (0, ∞) too. A continuous function f defined on I is called an operator convex function on I if f (s A + (1 − s) B )  sf ( A ) + (1 − s) f ( B ) for every 0 < s < 1 and for every pair of bounded self-adjoint operators A and B whose spectra are both in I . An operator concave function is likewise defined. If I = (0, ∞), then f (t ) is operator monotone on I if and only if f (t ) is operator concave and f (∞) > −∞ ([14], cf. [5]). This implies that every operator monotone function on (0, ∞) is operator concave. Then the associated operator mean A σ B is defined and represented as 1



1

1



1

A σ B = A 2 f A− 2 B A− 2 A 2

(1.1)

if A is invertible [7]. σ is said to be symmetric if A σ B = B σ A for every A, B. σ is symmetric if and only if f (t ) = t f (1/t ). When f (t ) = t a (0 < a < 1), the associated mean is denoted by A a B and called weighted geometric mean. In particular, ✩

*

This work was supported by JSPS KAKENHI Grant Number 21540181. This work was supported by Toyo University Inoue Enryo Kinen Kenkyuu. Corresponding author. E-mail addresses: [email protected] (M. Uchiyama), [email protected] (T. Yamazaki).

0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.11.055

M. Uchiyama, T. Yamazaki / J. Math. Anal. Appl. 413 (2014) 422–429

423

the case of a = 12 is the usual geometric mean and simply denoted by A  B. The arithmetic mean ∇ and the harmonic mean ! are naturally defined. It is well-known that A ! B  A  B  A ∇ B for every A , B  0; of course these are symmetric. It is well-known that 0 < A  B implies that B −1  A  A −1  A = I , but the converse does not hold. In the recent years, geometric means of n-matrices are studied by many authors. Let Pm be the set all m-by-m positive of n definite matrices. Define ω = ( w 1 , . . . , w n ) be a probability vector, i.e., w i > 0 for i = 1, . . . , n and i =1 w i = 1. Let n be the set of all probability vectors. For ω = ( w 1 , . . . , w n ) ∈ n , the Karcher mean Λ(ω; A 1 , . . . , A n ) of A 1 , . . . , A n ∈ Pm is characterized as the unique positive definite solution of the matrix equation [12] n 



1

1



w i log X − 2 A i X − 2 = 0.

i =1

If ω = ( n1 , . . . , n1 ) ∈ n , then the Karcher mean is simply written by Λ( A 1 , . . . , A n ). In the two matrices case, A , B ∈ Pm , the Karcher mean coincides with the weighted geometric mean. We note that the above matrix equation is called nthe Karcher equation [6]. The Karcher mean inherits many properties of geometric means (see [2,12,9,3]). For instance, i =1 w i A i  I implies Λ(ω; A 1 , . . . , A n )  I for ω = ( w 1 , . . . , w n ) ∈ n in [11,16]. Related to the Karcher mean, the power mean is also discussed in [10]. The power mean of n-matrices is inspired from the power mean of positive numbers. For t ∈ [−1, 1] \ {0} and ω = ( w 1 , . . . , w n ) ∈ n , the power mean P t (ω; A 1 , . . . , A n ) of A 1 , . . . , A n ∈ Pm is defined as the unique positive definite solution of the matrix equation n 

w i ( X t A i ) = X .

(1.2)

i =1

If ω = ( n1 , . . . , n1 ) ∈ n , then the power mean is simply written by P t ( A 1 , . . . , A n ). It is shown in [10] that the power mean of two matrices, A , B ∈ Pm , coincides with 1





1

1

P t (1 − w , w ; A , B ) = A 2 (1 − w ) I + w A − 2 B A − 2

t  1t

1

A2.

The power mean interpolates among the arithmetic, Karcher (geometric) and harmonic means. More precisely, the Karcher mean can be considered as the limit point of the power mean as t → 0, it is the same situation to the number case. We will introduce the details of relations among these means in Section 3. The aim of this paper is to investigate the converse of Loewner–Heinz inequality in the view point of operator mean. It is organized as follows: In Section 2, we shall show that if f (λ B + I )−1  f (λ A + I )  I for all sufficiently small λ  0, then f (λ A + I )  f (λ B + I ) and A  B. Moreover, we will deal with a symmetric operator mean. In Section 3, we will extend the results obtained in Section 2 in the case of the power means and the Karcher mean. 2. Operator inequality and operator mean We begin by recalling a few results which we will need later. If A  B  I , then A p  B p  I for all p  1 [1]. Actually, A  B p is decreasing for p  1 if A  B  I (see Corollary 3.3 of [13]). The following well-known result for positive invertible operators is essential (see [4]): p

log A  log B

⇐⇒

B −p  A p  I

for all p  0.

(2.1)

In this paper we deal with a non-constant operator monotone function f (t ) defined on a neighborhood of t = t 0 . However we assume t 0 = 1 for simplicity. In this case, for every bounded self-adjoint operator A the function f (λ A + I ) is well-defined for sufficiently small λ. We also note that f  (1) > 0. Theorem 1. Let f (t ) > 0 be a non-constant operator monotone function defined on a neighborhood of t = 1 with f (1) = 1, and let A and B be bounded self-adjoint operators. Then the following are equivalent: (i) (ii) (iii) (iv)

A  B, λ A + I  λ B + I for every λ  0, f (λ A + I )  f (λ B + I ) for all sufficiently small λ  0, f (λ B + I )−1  f (λ A + I )  I for all sufficiently small λ  0.

Proof. The equivalence of (i), (ii) and (iii) were shown in [15]. Since f (λ B + I ) is positive and invertible for sufficiently small λ  0, by (2.1), (iii) gives (iv). Assume (iv). For an arbitrary p > 0 take λ so that 0 < λ  p. Then as mentioned above we have p

p

f (λ B + I )− λ  f (λ A + I ) λ  I , because p /λ  1. We now show the next claim:

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lim  f (λ A + I )1/λ − exp f  (1) A  = 0.

(2.2)

λ→0

 Indeed, f (λt + 1)1/λ converges to e f (1)t uniformly for t in a bounded closed interval as λ → 0. Notice a := f  (1) > 0. From this claim it follows that

e − paB  e pa A  I for all p > 0. (2.1) yields a A = log ea A  log eaB = aB and hence (i).

2

We remark that (iv) is equivalent to f (λ A + I )−1  f (λ B + I )  I and that (i) follows (iv), provided (iv) holds for infinitely many λn > 0 that converge to 0. We now refer to the typical operator monotone functions t a and log t on (0, ∞). Corollary 2. Let A and B be self-adjoint operators, and let 0 < a  1. Then the following are equivalent: (i) (ii) (iii) (iv) (v)

A  B, λ A + I  λ B + I for every λ  0, (λ A + I )a  (λ B + I )a for all sufficiently small λ  0, log(λ A + I )  log(λ B + I ) for all sufficiently small λ  0, (λ B + I )−a  (λ A + I )a  I for all sufficiently small λ  0.

Proof. The implications

(i) −→ (ii) −→ (iii) −→ (iv) −→ (v) are trivial: indeed, the last one follows from (2.1). (v) −→ (i) follows from Theorem 1. However we give another direct proof. (v) implies ap

ap

(λ B + I )− λ  (λ A + I ) λ  I 1

for all 0 < λ  p. By the well-known formula limλ→0 (λ A + I ) λ = e A , we have

e −ap B  eap A  I for all p > 0. Hence we have a A = log ea A  log eaB = aB by (2.1).

2

We will often use the following well-known lemma later. Lemma 3. For positive invertible operators A 1 , . . . , A n and ω = ( w 1 , . . . , w n ) ∈ n ,

 lim

p 0

n 

 1p p w i Ai

 = exp

i =1

n 

 w i log A i ,

i =1

n

uniformly, i.e., (

i =1

p

n

1

w i A i ) p − exp(

i =1

w i log A i ) → 0 as p 0.

Theorem 4. Let f (t ) be an operator monotone function on (0, ∞) with f (1) = 1, and let A and B be bounded self-adjoint operators. Let σ be an operator mean satisfying !  σ  ∇ . Then A  B if and only if f (λ A + I ) σ f (−λ B + I )  I for all sufficiently small λ  0. Proof. Assume A  B. Since

I f

(λ A + I )+(−λ B + I ) 2

(λ A + I ) + (−λ B + I ) 2



 I holds for every positive number λ, we have

f (λ A + I ) + f (−λ B + I ) 2

= f (λ A + I ) ∇ f (−λ B + I )  f (λ A + I ) σ f (−λ B + I ), where the second inequality is due to the operator concavity of f . Assume conversely f (λ A + I ) σ f (−λ B + I )  I . By the λ

assumption we have f (λ A + I ) ! f (−λ B + I )  I . Since t p is operator concave for 0 < λ  p, we observe



p

p

f (λ A + I )− λ + f (−λ B + I )− λ 2

and then

− λp



f (λ A + I )−1 + f (−λ B + I )−1 2

−1

 I,

M. Uchiyama, T. Yamazaki / J. Math. Anal. Appl. 413 (2014) 422–429



p

p

f (λ A + I )− λ + f (−λ B + I )− λ

− 1p

425

 I.

2 In virtue of (2.2), we obtain



− 1p   e − f (1 ) p A + e f (1 ) p B

 I as λ → 0.

2

f  (1) (A 2

Letting p → 0, by Lemma 3, it yields exp(

− B ))  I . This implies A  B. 2

We remark that a symmetric operator mean σ , that is A σ B = B σ A for every A and B, satisfies !  σ  ∇ . The following corollary may make the above statement clearer. Corollary 5. Let f (t ) and h(t ) be operator monotone functions on (0, ∞) with f (1) = h(1) = 1, and let A and B be bounded selfadjoint operators. Let σ and δ be operator means satisfying !  σ  ∇ and !  δ  ∇ . Then h(λ A + I ) δ h(λ B + I )  I for all sufficiently small λ  0 if and only if f (λ A + I ) σ f (λ B + I )  I for all sufficiently small λ  0. Theorem 6. Let f (t ) be a non-constant operator monotone function on (0, ∞) with f (1) = 1, and let A and B be bounded self-adjoint operators. Then the following are equivalent: (i) A  B,

1

1

(ii) x 2  f (λ A + I )− 2 x

f (−λ B + I )− 2 x for all x ∈ H and all sufficiently small λ  0, (iii) x 2  e − p A x

e p B x for all x ∈ H and all p  0. To prove Theorem 6, we need Lemma 7. Let S 1 , . . . , S n be operators on H. Then the following are mutually equivalent:

n

(i) I  n1 i =1 t i S ∗i S i for all t 1 , . . . , tn > 0 with n (ii) x n  i =1 S i x for all x ∈ H.

n

i =1 t i

= 1,

Proof. Assume (i). Notice that each S i is non-singular: indeed, if S i x = 0 for a vector x ∈ H, then there is a {t i }ni=1 such that n  ti i =1

and

n

n

i =1 t i

S ∗i S i x, x < x, x

= 1. Since n  ti

x, x 

i =1

n

S ∗i S i x, x

for all x ∈ H, by putting t i as

n ti =

∗

j =1  S j S j x, x

1 n

 S ∗i S i x, x

,

n  ti

we have

x, x 

i =1

n

S ∗i S i x, x =

n 

2

S i x n .

i =1

We consequently get (ii). Next assume (ii). For t 1 , . . . , tn > 0 with

x 2 

n  i =1

This yields (i).

2

S i x n =

n  i =1

1



n1

t in S ∗i S i x, x



n  ti i =1

n

n

i =1 t i

= 1, we have

S ∗i S i x, x .

2

Proof of Theorem 6. By Theorem 4, A  B is equivalent to f (λ A + I )  f (−λ B + I )  I for all sufficiently small λ  0. Then we have

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 1 I  f (λ A + I )  f (−λ B + I ) = t f (λ A + I )  f (−λ B + I ) t

 1  f (−λ B + I )  t f (λ A + I ) ! 

t

for all t > 0, and obtain

I

f (λ A + I )−1 + t f (−λ B + I )−1

1 t

2

for all t > 0. By Lemma 7, we have (ii). Next we assume (ii). By Lemma 7

I

 

f (λ A + I )−1 + t f (−λ B + I )−1

1 t

2

{ 1t

f (λ A +

p

I )−1 } λ

p

+ {t f (−λ B + I )−1 } λ

 λp

2 λ

for all 0 < λ  p and all t > 0, where the last inequality follows from operator concavity of t p for λ/ p ∈ [0, 1]. Then we have p

I

p

p

p

( 1t ) λ f (λ A + I )− λ + t λ f (−λ B + I )− λ 2

.

It is equivalent to

 p  p 

x 2   f (λ A + I )− 2λ x f (−λ B + I )− 2λ x for all 0 < λ  p and x ∈ H by Lemma 7. Letting λ → 0, we have (iii) by (2.2) and replacing prove (iii) −→ (i). By Lemma 7, (iii) implies

I

e −2p A + e 2p B 2

p f  (1) 2

into p. Lastly, we will

,

and then

I

e −2p A + e 2p B

1p

2

for all p > 0. By Lemma 3, we have

I  exp

log e −2 A + log e 2B

This implies A  B.

2

= exp( B − A ).

2

Corollary 8. Let A and B be positive invertible operators. Then log A  log B if and only if x 2  A − p x

B p x for all p  0 and all x ∈ H. Corollary 8 has been already shown in [17] in the case of A = | T ∗ | and B = | T | (i.e., T is log-hyponormal). 3. Karcher and power means of multi-variable matrices We will discuss about the Karcher mean in this section. The Karcher mean is an extension of geometric mean of two matrices. So we will obtain extensions mentioned in the previous section. Moreover the Karcher mean can be considered as a limit of the power mean, and we shall consider more general results. In this section, we will discuss about only m-by-m matrices, hence H means Cm . Before stating our discussion, we shall introduce some properties of power mean for the reader’s convenience. Let ω = ( w 1 , . . . , w n ) ∈ n and A 1 , . . . , An ∈ Pm . By the definition of power mean (1.2), we have

P 1 (ω; A 1 , . . . , A n ) =

n  i =1

for t ∈ (0, 1]; especially

w i Ai

and

P t (ω; A 1 , . . . , A n ) = P −t



1 −1 ω; A − 1 , . . . , An

− 1

M. Uchiyama, T. Yamazaki / J. Math. Anal. Appl. 413 (2014) 422–429

 P −1 (ω; A 1 , . . . , A n ) =

427

− 1

n 

−1

w i Ai

.

i =1

Moreover, we have Lemma 9. (See [8,10,11].) The power mean P t (ω; A 1 , . . . , A n ) is increasing for t ∈ [−1, 1] \ {0}, and

lim P t (ω; A 1 , . . . , A n ) = Λ(ω; A 1 , . . . , A n ).

t →0

Henceforth, we use the symbol P 0 (ω; A 1 , . . . , A n ) instead of Λ(ω; A 1 , . . . , A n ). Theorem 10. Let A 1 , . . . , A n be Hermitian matrices, and ω = ( w 1 , . . . , w n ) ∈ n . Let f (t ) be a non-constant operator monotone function on (0, ∞) with f (1) = 1. Then the following are equivalent:

n

(i) i =1 w i A i  0, n (ii) P 1 (ω; f (λ A 1 + I ), . . . , f (λ A n + I )) = i =1 w i f (λ A i + I )  I for all sufficiently small λ  0, (iii) for each t ∈ [−1, 1], P t (ω; f (λ A 1 + I ), . . . , f (λ A n + I ))  I for all sufficiently small λ  0. Proof. Proof of (i) −→ (ii). It is obvious that (i) implies function with f (1) = 1, we have



I = f (I )  f

n 



w i (λ A i + I ) 

i =1

n 

n

i =1

w i (λ A i + I )  I for all λ  0. Since f (t ) is an operator concave

w i f (λ A i + I ).

i =1

(ii) −→ (iii) is given by only using Lemma 9, that is, Pt









ω; f (λ A 1 + I ), . . . , f (λ An + I )  P 1 ω; f (λ A 1 + I ), . . . , f (λ An + I ) =

n 

w i f (λ A i + I )  I .

i =1

We shall prove (iii) −→ (i). By Lemma 9, we have



n 

− 1 −1

w i f (λ A i + I )

   P t ω; f (λ A 1 + I ), . . . , f (λ An + I )  I .

i =1

Then we have

I



n 

−1

w i f (λ A i + I )



i =1

n 

w i f (λ A i + I )

i =1

for 0 < λ  p. Hence we have

 I

n 

p

w i f (λ A i + I )− λ

 1p .

i =1

By (2.2), we have

 I

n 

wie

− p f  (1 ) A i

 1p as λ → 0.

i =1

By Lemma 3, we have

 I  exp

n  i =1

that is, (i).

2

 w i log e

− f  (1 ) A i

,

− λp

 λp

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M. Uchiyama, T. Yamazaki / J. Math. Anal. Appl. 413 (2014) 422–429

Corollary 11. Let A 1 , . . . , A n be Hermitian matrices, and ω = ( w 1 , . . . , w n ) ∈ n . Let f (t ) be a non-constant operator monotone function on (0, ∞) with f (1) = 1. Then the following are equivalent:

n

w A  0, (i) ni =1 i i (ii) i =1 w i f (λ A i + I )  I for all sufficiently small λ  0, (iii) Λ(ω; f (λ A 1 + I ), . . . , f (λ A n + I ))  I for all sufficiently small λ  0. Proof. Since limt →0 P t (ω; A 1 , . . . , A n ) = Λ(ω; A 1 , . . . , A n ), we have (i) −→ (ii) −→ (iii). So we have only to prove (iii) −→ (i). It is known that

 p p Λ(ω; A 1 , . . . , An )  I implies Λ ω; A 1 , . . . , An  I for all p  1 [16]. Using this fact, (iii) implies

 p p Λ ω; f (λ A 1 + I ) λ , . . . , f (λ An + I ) λ  I

for 0 < λ  p. Letting λ → 0, we have

    Λ ω; e p f (1) A 1 , . . . , e p f (1) An  I for all p  0.

(3.1)

Here we note that it is shown in [16,11] that n 

w i log A i  0

⇐⇒

 p p Λ ω; A 1 , . . . , An  I for all p  0.

i =1

Hence, (3.1) is equivalent to n 

 w i log e f (1) A i  0.

i =1

Therefore we obtain (i).

2

We especially consider the probability vector

ω = ( n1 , . . . , n1 ) to obtain a multi-variable case of Theorem 6.

Theorem 12. Let A 1 , . . . , A n be Hermitian matrices, and let f be a non-constant operator monotone function on (0, ∞) with f (1) = 1. Then the following are equivalent: (i)

n

i =1

A i  0,

n

1

(ii) x n  i =1 f (λ A i + I )− 2 x for all sufficiently small λ  0 and all x ∈ H, n (iii) x n  i =1 e − p A i x for all x ∈ H and all p  0. Proof. Assume (i). We have

  Λ f (λ A 1 + I ), . . . , f (λ An + I )  I for all sufficiently small λ  0 by Corollary 11. Let t 1 , . . . , tn be positive numbers satisfying geometric means inequality, we have





I  Λ f (λ A 1 + I ), . . . , f (λ A n + I )



−1

−1

= Λ t 1 f (λ A 1 + I ), . . . , tn

 f (λ A n + I ) 



n  ti i =1

n

− 1 −1

f (λ A i + I )

that is,

I

n  ti i =1

n

f (λ A i + I )−1 .

Hence we have (ii) by Lemma 7. We next assume (ii). By Lemma 7, we have

1 n

I

n

i =1

 −1

t i f (λ A i + I )



p 1  − λp t i f (λ A i + I )− λ n

n

i =1

 λp

,

n

i =1 t i

= 1. Using harmonic–

M. Uchiyama, T. Yamazaki / J. Math. Anal. Appl. 413 (2014) 422–429

429

for all 0 < λ  p. Then p 1  − λp t i f (λ A i + I )− λ  I , n

n

i =1

and by Lemma 7, we obtain

x n 

n   p   f (λ A i + I )− 2λ x i =1

holds for all x ∈ H. Letting λ → 0, we have

x n 

n   − p f  (1) A  e 2 i x i =1

holds for all p > 0 by (2.2). Replacing p f  (1)/2 into p > 0, we have (iii). Lastly we assume (iii). By Lemma 7, we have

1  −p Ai e  I, n n

i =1

and we obtain



1  −p Ai e n n

 1p I

i =1

for all p > 0. Hence by Lemma 3, we have (i).

2

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