Inequalities on weighted classical pythagorean means ...

Ploymukda et al. / Malaysian Journal of Fundamental and Applied Sciences Vol. 16, No. 2 (2020) 223-227

223

Inequalities on weighted classical pythagorean means, Tracy-Singh products, and Khatri-Rao products for hermitian operators Arnon Ploymukda, Pattrawut Chansangiam*

Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand * Corresponding author: [email protected]

Article history Received 18 December 2018 Revised 10 September 2019 Accepted 26 September 2019 Published Online 15 April 2020

Abstract We establish a number of operator inequalities between three kinds of means, namely, weighted arithmetic/harmonic/geometric means, and two kinds of operator products, namely, Tracy-Singh products and Khatri-Rao products. In this study, we have validated the data under certain assumptions relying on (opposite) synchronization, comparability, and spectra of operators. The tensor product of operators, and Tracy-Singh/Khatri-Rao products of matrices as special cases are presented. Keywords: weighted arithmetic mean, weighted geometric mean, weighted harmonic mean, Tracy-Singh product, Khatri-Rao product

© 2020 Penerbit UTM Press. All rights reserved

INTRODUCTION

Throughout this paper, and denote the complex separable Hilbert spaces. When and are Hilbert spaces, let be the algebra of all bounded linear operators from into , and abbreviate

to . Denoted by the cone of positive operators on For any Hermitian operators and on the partial order

indicates that Two Hermitian operators and are comparable if or The expressions and

have the same meaning so that is both positive and invertible.

This paper focuses on the three classical Pythagorean means, namely, the arithmetic mean, the geometric mean, and the harmonic mean. Over the years, theory of these kinds of means for matrices and operators are significantly developed, see e.g. [1-3] and references therein. Recall that for any the -weighted arithmetic mean of is defined by

The -weighted harmonic mean of is defined by

In general, for any , we define (1)

Here, the limit is taken in the strong-operator topology. The -weighted geometric mean of two operators is defined by

For arbitrary positive operators, we define their weighted geometric mean by using the continuity argument as that for (1). In brief, we write

and for and , respectively.

Weighted classical Pythagorean means have the following remarkable properties where for any and

we have (2)

(3) It is well-known that for any we have

Here, denotes the tensor product. Recently, the theory of tensor product for operators is extended to that of Tracy-Singh product and Khatri-Rao product for operators, see e.g. [4-7]. The identity shown was generalized to that for weighted geometric means and Tracy-Singh products as follows: Proposition 1 ([8]). For any and we have

(4) In this paper, we establish further inequalities between three kinds

of weighted Pythagorean means, namely, weighted arithmetic/ harmonic/geometric means, and two kinds of operator products, namely, Tracy-Singh product and Khatri-Rao product. Our results include tensor product of operators, and Tracy-Singh/Khatri-Rao products of matrices as special cases.

The outline for the rest of paper is as follows. In Section 2, we present the preliminary results on Tracy-Singh product and Khatri-Rao product of Hilbert space operators. Section 3 begins with introducing the (opposite) synchronization between two ordered pairs of Hermitian operators. Then, we establish operator inequalities involving Tracy-Singh products and weighted arithmetic/harmonic means under the assumptions that two pairs of operators are (opposite) synchronous. In Section 4, we prove certain operator inequalities concerning Khatri-Rao products and weighted arithmetic/harmonic/ geometric means under suitable assumptions about synchronization, comparability, and spectra of operators.

H KX Y ( , )X YB

X Y

( , )X XB ( )XB ( )+HB.H. A B ,H

A BÖ ( ) .A B +- Î HB AB A BÖ .A BÑ 0A >

( )A ++Î HB A

[0,1],tÎ t, ( )A BÎ HB

(1 ) .tA B t A tB= - +"

t , ( )A B ++Î HB11 1! (1 ) .tA B t A tB-- -é ù= - +ë û

, ( )A B +Î HB

0! lim( )! ( ).t tA B A I B I

ee e

+®= + +

t, ( )A B +Î HB

( )1/2 1/2 1/2 1/2# .t

tA B A A BA A- -=

, !A B A B" #A B 1/2 1/2, !A B A B" 1/2#A B

, ( ) ,A B +Î HB ( ),X Î HB[0,1],tÎ

1 ,t tA B B As s -=* * *( ) ( ) ( ),t tX A B X X AX X BXs sÑ

1 1, ,A B 2 2, ( ) ,A B +ÎB H

1 1 2 2 1 2 1 2( # ) ( # ) ( )#( ).A B A B A A B BÄ = Ä ÄÄ

1 1 2, , ,A B A 2 ( )B +Î HB [0,1],tÎ

1 1 2 2 1 2 1 2( # ) ( # ) ( )# ( ).t t tA B A B A A B B=) ) )

RESEARCH ARTICLE

Muther et al. / Malaysian Journal of Fundamental and Applied Sciences Vol. 16, No. 2 (2020) 223-227

224

PRELIMINARIES ON TRACY-SINGH PRODUCT AND KHATRI-RAO PRODUCT OF HILBERT SPACE OPERATORS

In order to define the Tracy-Singh product for Hilbert space operators, we have to fix the following orthogonal decompositions:

(5)

where and are Hilbert spaces for Thus, each operator and can be uniquely represented as operator

matrices

and (6)

where and for each and

Definition 2. Let and be operator matrices in and respectively. We define the Tracy-Singh product

of and to be the bounded linear operator from

into itself, represented by

(7)

Lemma 3 ([4]). Let be compatible operators. Then (i) (ii) for any (iii)

(iv) If and are invertible, then (v) If and then

To define the Khatri-Rao product of operators, we fix the

decomposition (5) and assume that

Definition 4. Let and The

Khatri-Rao product of and is defined to be the bounded linear operator from into itself, represented by

(8)

Lemma 5. ([5]). There exists a bounded linear operator such that

(9) for any and The operator in Lemma 5 is called the selection operator associated with the ordered tuple

OPERATOR INEQUALITIES ON WEIGHTED CLASSICAL PYTHAGOREAN MEANS AND TRACY-SINGH PRODUCTS

In this section, we establish operator inequalities involving weighted arithmetic/harmonic means and Tracy-Singh products under the assumptions that two pairs of operators are (opposite) synchronous. Definition 6. Two ordered pairs and of Hermitian operators are said to be synchronous if either

for or for The pairs and are said to be opposite synchronous if either

and or and

Now, we establish operator inequalities involving weighted harmonic means and Tracy-Singh products. Theorem 7. Let and (i) If and are synchronous, then

(10) (ii) If and are opposite synchronous, then

(11) Proof. (i) First, suppose The case

leads to and A2-1,, B2

-1. Thus, by Lemma 3 we have (12)

The case also leads to the inequality (12). It follows from (12) that

Thus

Hence

For arbitrary perturb each of them with and then take limit as (ii) By continuity, we may assume that If and are opposite synchronous, then we get the reverse of (12). Hence, in this case, we get (11). Corollary 8. If two positive operators and are comparable, then for any

(13) (14)

(15)

(16) Here, in (15) and (16), we assume further that and are invertible. Proof. First, suppose Then the pairs and are synchronous. By Theorem 7, we get (13). Since and are opposite synchronous, Theorem 7 yields the inequality (14). The synchronization between and implies the inequality (15). The opposite synchronization between and implies the inequality (16). The case can be similarly treated.

=1 =1= , =

m n

i ki kÅ ÅH H K K

iH_ jK , .i j

( )AÎ HB ( )BÎ KB

,

, =1=

m m

ij i jA Aé ùë û [ ] ,, =1= n n

kl k lB B

( , )ij j iA Î H HB ( , )kl l kB Î K KB , = 1, ,i j m

, = 1, , .k l n

,, 1[ ]m mij i jA A == ,

, 1[ ]n nkl k lB B ==( )HB ( ),KB

A B ,

, =1

m ni ki k ÄÅ H K

.ij kl kl ijA B A Bé ùé ù= Äë ûë û)

1 1 2 2, , ,A B A B

1 1 2 2 1 2 1 2 1 2 1 2( ) ( ) .A B A B A A A B B A B B+ + = + + +) ) ) ) )

1 2 1 2 1 2( ) ( ) ( )A A A A A Aa a a= =) ) ) .a Î

1 2 1 2 1 1 2 2( )( ) ( ) ( ).A A B B AB A B=) ) )

1A 2A 1 1 11 2 1 2( ) .A A A A- - -=) )

1 2 0A AÖ Ö 1 2 0,B BÖ Ö 1 1 2 2 0.A B A B) )Ö Ö

.m n=

,

, =1= ( )

n n

ij i jA Aé ù Îë û HB

,

, =1= ( ).

n n

ij i jB Bé ù Îë û KB

A B

=1

ni ii ÄÅ H K

,

, 1.

n n

ij ij i jA B A B

=é ù= Äë û*

Z*( )A B Z A B Z=* )( )AÎ HB ( ).BÎ KB

Z( , ).H K

1 2( , )A A 1 2( , )B B

i iA BÑ 1,2,i = i iA BÖ 1,2.i =

1 2( , )A A 1 2( , )B B

1 1A BÑ 2 2 ,A BÖ 1 1A BÖ 2 2.A BÑ

1 1 2 2, , , ( )A B A B +Î HB [0,1].tÎ

1 2( , )A A 1 2( , )B B

1 1 2 2 1 2 1 2( ! ) ( ! ) ( ) ! ( ). t t tA B A B A A B B) ) )Ö

1 2( , )A A 1 2( , )B B

1 1 2 2 1 2 1 2( ! ) ( ! ) ( ) ! ( ).t t tA B A B A A B B) ) )Ñ

1 1 2 2, , , 0.A B A B > 1 1 2 2,A B A BÖ Ö1 11 1A B- -Ñ

1 1 1 11 1 2 2( ) ( ) 0.A B A B- - - -- -) Ö

1 1 2 2,A B A BÑ Ñ

1 1 1 1 1 11 2 1 2 1 20 (1 ) (1 ) (1 )t t A A t t B B t t A B- - - - - -- + - - -) ) )Ñ1 11 2(1 )t t B A- -- - )

2 1 1 2 1 11 2 1 2(1 ) (1 ) ( )t t A A t t B B- - - -é ù= - - - + -ë û ) )

1 1 1 11 2 1 2(1 ) (1 )t t A B t t B A- - - -- - - -) )

1 1 1 11 2 1 2(1 )t A A tB B- - - -= - +) )

1 1 1 11 1 2 2(1 ) (1 ) .t A tB t A tB- - - -é ù é ù- - + - +ë û ë û)

1 1 1 11 1 2 2(1 ) (1 )t A tB t A tB- - - -é ù é ù- + - +ë û ë û)

1 1 1 11 2 1 2(1 ) .t A A tB B- - - -- +) )Ñ

1 2 1 2( ) ! ( )tA A B B) )

{ } 11 11 2 1 2(1 )( ) ( )t A A t B B

-- -= - +) )

{ } 11 1 1 11 2 1 2(1 )t A A tB B

-- - - -= - +) )

{ } 11 1 1 11 1 2 2(1 ) (1 )t A tB t A tB

-- - - -é ù é ù- + - +ë û ë û)Ñ

1 11 1 1 11 1 2 2(1 ) (1 )t A tB t A tB

- -- - - -é ù é ù= - + - +ë û ë û)

1 1 2 2( ! ) ( ! ).t tA B A B= )

1 1 2 2, , , 0,A B A B Ö Ie

0 .e +®

1 1 2 2, , , 0.A B A B > 1 2( , )A A

1 2( , )B B

A B[0,1],tÎ

( ! ) ( ! ) ( ) ! ( ),t t tA B A B A A B B) ) )Ö( ! ) ( ! ) ( ) ! ( ),t t tA B B A A B B A) ) )Ñ

1 1 1 1( ! ) ( ! ) ( )! ( ),t t tA B B A A B B A- - - -) ) )Ö1 1 1 1( ! ) ( ! ) ( )! ( ).t t tA B A B A A B B- - - -) ) )Ñ

A B

.A BÑ ( , )A A ( , )B B( , )A B ( , )B A

1( , )A B- 1( , )B A-1( , )A A- 1( , )B B-

A BÖ


225

Next, we discuss relations between weighted arithmetic means and Tracy-Singh products in terms of inequalities. Theorem 9. Let be Hermitian operators and

(i) If and are synchronous, then

(17) (ii) If and are opposite synchronous, the

(18) Proof. (i) Since and are synchronous, by Lemma 3 we have By using Lemma 3 again, we obtain

Thus (ii) For the opposite synchronous case, we have

and hence the inequality (18) holds. Corollary 10. If two Hermitian operators and are comparable, then for any

(19) (20)

(21)

(22) Here, in (21) and (22), we assume further that and are invertible. Proof. The proof is similar to that of Corollary 8. OPERATOR INEQUALITIES ON WEIGHTED CLASSICAL PYTHAGOREAN MEANS AND KHATRI-RAO PRODUCTS

In this section, we present a number of operator inequalities involving Khatri-Rao products and weighted arithmetic/harmonic/ geometric means under suitable assumptions about synchronization, comparability, and spectra of operators. Corollary 11. Let be comparable operators. If

then for any

Proof. By making use of Lemma 5, property (3), Corollary 10 and property (2), we obtain

Theorem 12. Let and If then

Proof. It follows Lemma 5, property (3), Proposition 1, and property (2) that

Corollary 13. For any such that is invertible, we have

Proof. Note that for any such that is invertible, we have It follows this fact and Theorem 12 with

that

Theorem 14. Let and If then

Proof. Recall that the Khatri-Rao product is continuous with respect to the operator norm (see [9]). By continuity of the Khatri-Rao product, the -weighted arithmetic mean and the -weighted harmonic mean, we may assume that and are invertible.

It follows this fact and Corollary 13 that

In [8], it was shown that for any in and

(23)

Now, we give a reverse inequality of (23). Recall that a linear map between two operator algebras is said to be positive if it preserves positive operators; is said to be unital if it preserves the identity operator. Lemma 15 ([10]). Let be such that and

where are positive constants. Denote and Then, for any positive linear map

and we have (24)

where

1 1 2 2, , , ( )A B A B Î HB[0,1].tÎ

1 2( , )A A 1 2( , )B B

1 1 2 2 1 2 1 2( ) ( ) ( ) ( ).t t tA B A B A A B B) ) )" " Ñ "

1 2( , )A A 1 2( , )B B

1 1 2 2 1 2 1 2( ) ( ) ( ) ( ).t t tA B A B A A B B) ) )" " Ö "

1 2( , )A A 1 2( , )B B

1 1 2 2( ) ( ) 0.A B A B- -) Ö

[ ]1 1 2 20 (1 ) ( ) ( )t t A B A B- - -)Ñ

[ ]1 2 1 2 1 2 1 2(1 )t t A A A B B A B B= - - - +) ) ) )

1 2 1 2[(1 )( ) ( )]t A A t B B= - +) )

1 1 2 2[(1 ) ] [(1 ) ]t A tB t A tB- - + - +)

1 2 1 2 1 1 2 2[( ) ( )] [( ) ( )].t t tA A B B A B A B= -) ) )" " "

1 1 2 2 1 2 1 2( ) ( ) ( ) ( ).t t tA B A B A A B B) ) )" " Ñ "

1 1 2 2( ) ( ) 0A B A B- -) Ñ

A B[0,1],tÎ

( ) ( ) ( ) ( ),t t tA B A B A A B B) ) )" " Ñ "( ) ( ) ( ) ( ),t t tA B B A A B B A) ) )" " Ö "

1 1 1 1( ) ( ) ( ) ( ),t t tA B B A A B B A- - - -) ) )" " Ñ "1 1 1 1( ) ( ) ( ) ( ).t t tA B A B A A B B- - - -) ) )" " Ö "

A B

(, )A B +Î HB A B =*,B A* [0,1],tÎ

1( ! ) ( ! ).t tA B A B A B-* *Ö

( ) ! ( )tA B A B B A=* * ** *( ) ! ( )tZ A B Z Z B A Zé ùë= ûû é ùë) )

[ ]* ( )! ( )tZ A B B A Z) )Ö

[ ]* ( ! ) ( ! )t tZ A B B A Z)Ö( ! ) ( ! )t tA B B A= * 1( ! ) ( ! ).t tA B A B-= *

(, )A B +Î HB (0,1).tÎ ,A B B A=* *

1( # ) ( # ).t tA B A B A B-* *Ö

( ) # ( )tA B A B B A=* * ** *( ) # ( )tZ A B Z Z B A Zé ù é ù= ë û ë û) )

[ ]* ( ) # ( )tZ A B B A Z) )Ö

[ ]* ( # ) ( # )t tZ A B B A Z= )( # ) ( # )t tA B B A= *

1( # ) ( # ).t tA B A B-= *

(, )X Y +Î HB Y

1( ) .Y XY X X X-* *Ö

(, )A B +Î HB A1#( ) .A BA B B- =

1 / 2t =1 1 1( ) ( # ) ( # ) .Y XY X Y XY X Y XY X X X- - - =* * *Ö

(, )A B +Î HB (0,1).tÎ ,A B B A=* *

1( ) ( ! ).t tA B A B A B-* *Ö "

t tA B

( ) ( )1 11 1 .X Y X X X Y X- -- -+ = - +

1( ) ( ! )t tA B A B-*"

[ ] 11 1(1 ) (1 )t A tB tA t B-- -= - + +é ùë û-*

[ ] [ ]{ }11(1 ) (1 ) (1 )t A tB t A t A t A tB A--= - + - - - +*

[ ] 1(1 )t A tB t A-= - + *

[ ] [ ]{ }11(1 ) (1 ) (1 )t A tB t t A t A tB A--- - + - - +*1(1 )( )t t A A B A-= - +* *

[ ] [ ]{ }11(1 ) (1 ) (1 )t t t A tB A t A tB A--- - - + - +*1 1(1 )( ) (1 )( )t t A A A B t t A A- -- + - -* * *Ñ

.A B= *

1 1 2 2, , ,A B A B ( )+HB[0,1],tÎ

1 1 2 2 1 2 1 2( # ) ( # ) ( ) # ( ).t t tA B A B A A B B* * *ÑF

F

(, )A B +Î HB 1 1m I A M IÑ Ñ

2 2m I B M IÑ Ñ 1 2 1 2, , ,m m M M

2 1/m m M= 2 1/ .M M m=( ) (: )F ®H HB B (0,1),tÎ

[ ]( # ) ( ) # ( )t tA B A BlF F FÖ

Muther et al. / Malaysian Journal of Fundamental and Applied Sciences Vol. 16, No. 2 (2020) 223-227

226

(25)

Theorem 16. Let be such that

and Let and Then for any we have

(26)

where is given by (25). Proof. Consider a map where is the selection operator described in Lemma 5, associated with the ordered tuple

Then, is a unital positive linear map. It follows Lemma 5, Proposition 1, and Lemma 15 that

Corollary 17. Let and Let be positive constants, denote and and define

as in (25). (i) If and then

(ii) If and then

Proof The assertion (i) follows from Theorem 16 by setting

and To prove (ii), set and in the same theorem and property (2).

Recall the following relation between a positive linear map and the

harmonic mean of operators: Lemma 18 ([10]). Let be such that and If is a positive linear map, then

Theorem 19. Let and (i) Suppose and If

and are synchronous, then

(27) where

(28)

(ii) If and are opposite synchronous, then (29)

Proof. (i) Assume that and are synchronous. The cases and are trivial. Now, By using Lemmas 5 and 18, and Theorem 12, we have

Assume that and are opposite synchronous. By applying Lemma 5, Proposition 7 and property (3), we obtain that for any

Corollary 20. Let and Assume that and

are comparable. Then (30)

(31)

Moreover, if and then

(32)

Here, the constant is given by (28). Proof. The results in this corollary are consequences of Theorem 4.9. The opposite synchronization between and leads to (30). The opposite synchronization between and implies

(31). If and

then by Lemma 3, we have and and the inequality (32) follows.

Theorem 21. Let be Hermitian and (i) If and are synchronous, then

(33) (ii) If and are opposite asynchronous, the

(34) Proof. (i) By using Lemma 5 and Theorem 9, we obtain

(ii) The proof is similar to that of (i). Corollary 22. If two Hermitian operators and are comparable, then

1 .(1 )( )

tt t t t

t t

Mm mM t M mt M m t Mm mM

læ ö- - -

= × ×ç ÷- - -è ø

1 1 2 2, , ( ),A B A B ++Î HB

1 1 2 10 m I A A M I< )Ñ Ñ 2 1 2 20 .m I B B M I< )Ñ Ñ

2 1/m m M= 2 1/ .M M m= 0 1,t< <

[ ]1 1 2 2 1 2 1 2( # ) ( # ) ( ) # ( )t t tA B A B A A B Bl* * *Öl

*: T Z TZF Z

( , ).H H F

1 1 2 2( # ) ( # )t tA B A B* [ ]*1 1 2 2( # ) ( # )t tZ A B A B Z= )

[ ]*1 2 1 2( ) # ( )tZ A A B B Z= ) )

{ }* *1 2 1 2( ) # ( )tZ A A Z Z B B Zl é ù é ùë û ë û) )Ö

[ ]1 2 1 2( ) # ( ) .tA A B Bl= * *

(, )A B ++Î HB (0,1).tÎ 1 2 1 2, , ,m m M M

2 1/m m M= 2 1/ ,M M m=l

1 1m I A A M I)Ñ Ñ 2 2 ,m I B B M I)Ñ Ñ( # ) ( # ) [( ) # ( )].t t tA B A B A A B Bl* * *Ö

1 1m I A B M I)Ñ Ñ 2 2 ,m I B A M I)Ñ Ñ

1( # ) ( # ) [( ) # ( )].t t tA B A B A B B Al-* * *Ö

1 2 :A A A= = 1 2 : .B B B= = 1 2 :A B A= =

1 2 :B A B= =

(, )A B ++Î HB 1 10 m I A M I< Ñ Ñ

2 20 .m I B M I< Ñ Ñ ( ) (: )F ®H HB B

( )[ ]

2

1 1 2 2

1 2 1 2

( ! ) ( )! ( ) .( )( )

M m M mA B A B

M m m M

+F F F

+ +Ö

1 1 2 2, , ( ),A B A B ++Î HB [0,1].tÎ

1 1 2 10 m I A A M I< )Ñ Ñ 2 1 2 20 .m I B B M I< )Ñ Ñ

1 2( , )A A 1 2( , )B B

[ ]21 1 2 2 1 2 1 2( ! ) ( ! ) ( )! ( )t t tA B A B k A A B B* * *Ö

1 1 1 2 2

1 1 2 1 1 2

( # ) ( # ) .( )#( )

t

t t

m M m Mkm M M m

-

- -

="

" "

1 2( , )A A 1 2( , )B B

1 1 2 2 1 2 1 2( ! ) ( ! ) ( ) ! ( ).t t tA B A B A A B B* * *Ñ

1 2( , )A A 1 2( , )B B0t = 1t = (0,1).tÎ

1 1 2 2( ! ) ( ! )t tA B A B*

[ ]*1 1 2 2( ! ) ( ! )t tZ A B A B Z= )

[ ]*1 2 1 2( )! ( )tZ A A B B Z) )Ö

*1 2 1 2

1 1( ) ! ( )1

Z A A B B Zt t

ì üé ù é ù= í ýê ú ê ú-ë û ë ûî þ) )

2

1 1 2 2

1 2 1 2

1

1 1

M m M mt t

M m m Mt t t t

æ ö+ç ÷ç ÷-è ø

æ öæ ö+ +ç ÷ç ÷- -è øè ø

Ö

* *1 2 1 2

1 1( ) ! ( )1

Z A A Z Z B B Zt t

ì üé ù é ù´í ýê ú ê ú-ë û ë ûî þ) )

[ ]2

1 1 1 2 21 2 1 2

1 1 2 1 1 2

( # ) ( # ) ( ) ! ( ) .( )#( )

tt

t t

m M m M A A B BM m m M

-

- -

æ ö= ç ÷è ø

* *"" "

1 2( , )A A 1 2( , )B B

[0,1],tÎ

1 1 2 2( ! ) ( ! )t tA B A B* [ ]*1 1 2 2( ! ) ( ! )t tZ A B A B Z= )

[ ]*1 2 1 2( )! ( )tZ A A B B Z) )Ñ

* *1 2 1 2( ) ! ( )tZ A A Z Z B B Zé ù é ùë û ë û) )Ñ

1 2 1 2( ) ! ( ).tA A B B* *Ñ

(, )A B ++Î HB [0,1].tÎ AB( ! ) ( ! ) ( ) ! ( ),t t tA B B A A B B A* * *Ñ

1 1 1 1( ! ) ( ! ) ( )! ( ).t t tA B A B A A B B- - - -* * *Ñ

1 10 m I A M I< Ñ Ñ 2 20 ,m I B M I< Ñ Ñ

[ ]2( ! ) ( ! ) ( )! ( ) .t t tA B A B k A A B B* * *Ök

( , )A B ( , )B A1( , )A A- 1( , )B B-

1 10 m I A M I< Ñ Ñ 2 20 ,m I B M I< Ñ Ñ

1 1m I A A M I)Ñ Ñ 2m I B B)Ñ

2 ,M IÑ

1 1 2 2, , , ( )A B A B Î HB [0,1].tÎ

1 2( , )A A 1 2( , )B B

1 1 2 2 1 2 1 2( ) ( ) ( ) ( ).t t tA B A B A A B B* * *" " Ñ "

1 2( , )A A 1 2( , )B B

1 1 2 2 1 2 1 2( ) ( ) ( ) ( ).t t tA B A B A A B B* * *" " Ö "

1 1 2 2( ) ( )t tA B A B*" " [ ]*1 1 2 2( ) ( )t tZ A B A B Z= )" "

[ ]*1 2 1 2( ) ( )tZ A A B B Z) )Ñ "* *

1 2 1 2(1 ) ( ) ( )t Z A A Z tZ B B Z= - +) )

1 2 1 2(1 )( ) ( )t A A t B B= - +* *

1 2 1 2( ) ( ).tA A B B= * *"

A B


227

(4.13) (4.14)

(4.15) (4.16)

Here, in (37) and (38), we assume further that and are invertible. Proof. The proof is similar to that of Corollary 8. CONCLUSIONS

We provide a number of operator inequalities between three classical Pythagorean means and two kinds of operator products, namely, Tracy-Singh products and Khatri-Rao products. Each inequality is valid under certain assumptions relying on (opposite) synchronization, comparability, and spectra of operators. Our results include tensor product of operators, and Tracy-Singh and Khatri-Rao products of matrices as special cases.

ACKNOWLEDGEMENT The first author expresses his gratitude towards Thailand Research Fund for providing the Royal Golden Jubilee PhD Scholarship, grant No. PHD60K0225 throughout his research.

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Jersey, 2007. [2] Hiai, F. 2010. Matrix analysis: matrix monotone functions, matrix means,

and majorizations, Interdisciplinary Information Sciences. 16(2), 139-248.

[3] Hiai, F. and Petz, D. Introduction to Matrix Analysis and Applications, Springer, New Delhi, 2014.

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