On reverses of the Golden-Thompson type inequalities
aa r X i v : . [ m a t h . F A ] A ug ON REVERSES OF THE GOLDEN-THOMPSON TYPEINEQUALITIES
MOHAMMAD BAGHER GHAEMI, VENUS KALEIBARY AND SHIGERU FURUICHI
Abstract.
In this paper we present some reverses of the Golden-Thompsontype inequalities: Let H and K be Hermitian matrices such that e s e H (cid:22) ols e K (cid:22) ols e t e H for some scalars s ≤ t , and α ∈ [0 , p > k = 1 , , . . . , nλ k ( e (1 − α ) H + αK ) ≤ (max { S ( e sp ) , S ( e tp ) } ) p λ k ( e pH ♯ α e pK ) p , where A♯ α B = A (cid:0) A − B A − (cid:1) α A is α -geometric mean, S ( t ) is the so calledSpecht’s ratio and (cid:22) ols is the so called Olson order. The same inequalities arealso provided with other constants. The obtained inequalities improve someknown results. introduction In what follows, capital letters
A, B, H and K stand for n × n matrices orbounded linear operators on an n -dimentional complex Hilbert space ( H , h · i ).For a pair A, B of Hermitian matrices, we say A ≤ B if B − A ≥
0. Let A and B be two positive definite matrices. For each α ∈ [0 , A♯ α B of A and B in the sense of Kubo-Ando [10] is defined by A♯ α B = A (cid:0) A − B A − (cid:1) α A . Also for positive definite matrices A and B , the weak log-majorization A ≺ wlog B means that k Y j =1 λ j ( A ) ≤ k Y j =1 λ j ( B ) , k = 1 , , · · · , n, where λ ( A ) ≥ λ ( A ) ≥ · · · ≥ λ n ( A ) are the eigenvalues of A listed in decreasingorder. If equality holds when k = n , we have the log-majorization A ≺ log B . It isknown that the weak log-majorization A ≺ wlog B implies k A k u ≤ k B k u for anyunitarily invariant norm k · k u , i.e. k U AV k u = k A k u for all A and all unitaries U, V . See [2] for theory of majorization.In [15], Specht obtained an inequality for the arithmetic and geometric meansof positive numbers: Let x ≥ . . . ≥ x n > t = x /x n . Then x + . . . + x n n ≤ S ( t )( x . . . x n ) n , Mathematics Subject Classification.
Primary 15A42; Secondary 15A60, 47A63.
Key words and phrases.
Ando-Hiai inequality, Golden-Thompson inequality, Eigenvalue in-equality, Geometric mean, Olson order, Specht ratio, Generalized Kantorovich constant, Uni-tarily invariant norm. where S ( t ) = ( t − t / ( t − e log t ( t = 1) and S (1) = 1 (1.1)is called the Specht ratio at t . Note that lim p → S ( t p ) p = 1, S ( t − ) = S ( t ) > t = 1 , t > T re H + K ≤ T re H e K for arbitrary Hermitian matrices H and K . This inequality has been comple-mented in several ways [1, 9]. Ando and Hiai in [1] proved that for every unitarilyinvariant norm k · k u and p > k ( e pH ♯ α e pK ) p k u ≤ k e (1 − α ) H + αK k u . (1.2)Seo in [13] found some upper bounds on k e (1 − α ) H + αK k u in terms of scalarmultiples of k ( e pH ♯ α e pK ) p k u , which show reverse of the Golden-Thompson typeinequality (1.2). In this paper we establish another reverses of this inequality,which improve and refine Seo’s results. In fact the general sandwich condition sA ≤ B ≤ tA for positive definite matrices, is the key for our statements. Also,the so called Olson order (cid:22) ols is used. For positive operators, A (cid:22) ols B if and onlyif A r ≤ B r for every r ≥ reverse inequalities via specht ratio To study the Golden-Thompson inequality, Ando-Hiai in [1] developed thefollowing log-majorizationes: A r ♯ α B r ≺ log ( A♯ α B ) r , r ≥ , or equivalently ( A p ♯ α B p ) p ≺ log ( A q ♯ α B q ) q , < q ≤ p. There are some literatures [14] on the converse of these inequalities in terms ofunitarily invariant norm k · k u . By the following lemmas, we obtain a new reverseof these inequalities in terms of eigenvalue inequalities. Lemma 2.1.
Let A and B be positive definite matrices such that sA ≤ B ≤ tA for some scalars < s ≤ t , and α ∈ [0 , . Then A r ♯ α B r ≤ (max { S ( s ) , S ( t ) } ) r ( A♯ α B ) r , < r ≤ , (2.1) where S ( t ) is the Specht’s ratio defined as (4.3) .Proof. Let f be an operator monotone function on [0 , ∞ ). Then according to theproof of Theorem 1 in [7], we have f ( A ) ♯ α f ( B ) ≤ f ( M ( A♯ α B )) , N REVERSES OF THE GOLDEN-THOMPSON TYPE INEQUALITIES 3 where M = max { S ( s ) , S ( t ) } . Putting f ( t ) = t r for 0 < r ≤
1, we reach inequality(2.1). (cid:3)
Lemma 2.2.
Let A and B be positive definite matrices such that sA (cid:22) ols B (cid:22) ols tA for some scalars < s ≤ t , and α ∈ [0 , . Then λ k ( A♯ α B ) r ≤ max { S ( s r ) , S ( t r ) } λ k ( A r ♯ α B r ) , r ≥ , (2.2) and hence, λ k ( A q ♯ α B q ) q ≤ (max { S ( s p ) , S ( t p ) } ) p λ k ( A p ♯ α B p ) p , < q ≤ p, (2.3) where S ( t ) is the Specht’s ratio defined as (4.3) and k = 1 , , . . . , n .Proof. First note that the condition sA (cid:22) ols B (cid:22) ols tA , is equivalent to thecondition s ν A ν ≤ B ν ≤ t ν A ν for every ν ≥
1. In particular, we have sA ≤ B ≤ tA for ν = 1. Also, for r ≥ < r ≤ A r ♯ α B r ≤ (max { S ( s ) , S ( t ) } ) r ( A♯ α B ) r . (2.4)On the other hand, from the condition s ν A ν ≤ B ν ≤ t ν A ν for every ν ≥ ν = r , we have s r A r ≤ B r ≤ t r A r . Now if we let X = A r , Y = B r , w = s r and z = t r , then wX ≤ Y ≤ zX. (2.5)Using (2.4) under the condition (2.5), we have X r ♯ α Y r ≤ (max { S ( w ) , S ( z ) } ) r ( X♯ α Y ) r , and this is the same as A♯ α B ≤ (max { S ( s r ) , S ( t r ) } ) r ( A r ♯ α B r ) r . Hence λ k ( A♯ α B ) ≤ (max { S ( s r ) , S ( t r ) } ) r λ k ( A r ♯ α B r ) r . By taking r -th power on both sides and using the Spectral Mapping Theorem, weget the desired inequality (2.2). Note that from the minimax characterization ofeigenvalues of a Hermitian matrix [2] it follows immediately that A ≤ B implies λ k ( A ) ≤ λ k ( B ) for each k . Similarly since p/q ≥
1, from inequality (2.2) λ k ( A♯ α B ) pq ≤ max { S ( s pq ) , S ( t pq ) } λ k ( A pq ♯ α B pq ) . (2.6)Replacing A and B by A q and B q in (2.6), and using the sandwich condition s q A q ≤ B q ≤ t q A q , we have λ k ( A q ♯ α B q ) pq ≤ max { S ( s p ) , S ( t p ) } λ k ( A p ♯ α B p ) . This completes the proof. (cid:3)
Note that eigenvalue inequalities immediately imply log-majorization and uni-tarily invariant norm inequalities.
M.B. GHAEMI, V. KALEIBARY AND S. FURUICHI
Corollary 2.3.
Let A and B be positive definite matrices such that mI ≤ A, B ≤ M I for some scalars < m ≤ M with h = M/m , and let α ∈ [0 , . Then A r ♯ α B r ≤ S ( h ) r ( A♯ α B ) r , < r ≤ , (2.7) and hence λ k ( A♯ α B ) r ≤ S ( h r ) λ k ( A r ♯ α B r ) , r ≥ , (2.8) λ k ( A q ♯ α B q ) q ≤ S ( h p ) p λ k ( A p ♯ α B p ) p , < q ≤ p. (2.9) where S ( t ) is the Specht’s ratio defined as (4.3) and k = 1 , , . . . , n .Proof. Since mI ≤ A, B ≤ M I implies mM A ≤ B ≤ Mm A , the inequality (2.7) isobtained by letting s = m/M , t = M/m in Lemma 2.1. Also from mI ≤ A, B ≤ M I , we have m ν I ≤ A ν , B ν ≤ M ν I for every ν ≥
1, and so( mM ) ν A ν ≤ B ν ≤ ( Mm ) ν A ν . (2.10)Using Lemma 2.2 under the condition (2.10), we reach inequalities (2.8) and (2.9).Note that S ( h ) = S ( h ). (cid:3) Remark . We remark that the matrix inequality (2.7) is more stronger thancorresponding norm inequality obtained by Seo in [13, Corollary 3.2]. Also in-equality (2.9) is presented in [13, Lemma 3.1].In the sequel we show a reverse of the Golden-Thompson type inequality (1.2),which is our main result.
Theorem 2.5.
Let H and K be Hermitian matrices such that e s e H (cid:22) ols e K (cid:22) ols e t e H for some scalars s ≤ t , and let α ∈ [0 , . Then for all p > , λ k ( e (1 − α ) H + αK ) ≤ (max { S ( e sp ) , S ( e tp ) } ) p λ k ( e pH ♯ α e pK ) p , where S ( t ) is the so called Specht’s ratio defined as (4.3) and k = 1 , , . . . , n .Proof. Replacing A and B by e H and e K in the inequality (2.3) of Lemma 2.2,we can write λ k ( e qH ♯ α e qK ) q ≤ (max { S ( e sp ) , S ( e tp ) } ) p λ k ( e pH ♯ α e pK ) p , < q ≤ p. By [9, Lemma 3.3], we have e (1 − α ) H + αK = lim q → ( e qH ♯ α e qK ) q , and hence it follows that for each p > λ k ( e (1 − α ) H + αK ) ≤ (max { S ( e sp ) , S ( e tp ) } ) p λ k ( e pH ♯ α e pK ) p . (cid:3) Corollary 2.6.
Let H and K be Hermitian matrices such that e s e H (cid:22) ols e K (cid:22) ols e t e H for some scalars s ≤ t , and let α ∈ [0 , . Then for every unitarily invariantnorm k · k u and all p > , k e (1 − α ) H + αK k u ≤ (max { S ( e sp ) , S ( e tp ) } ) p k ( e pH ♯ α e pK ) p k u , (2.11) N REVERSES OF THE GOLDEN-THOMPSON TYPE INEQUALITIES 5 and the right-hand side of (2.11) converges to the left-hand side as p ↓ . Inparticular, k e H + K k u ≤ max { S ( e s ) , S ( e t ) }k ( e H ♯e K ) k u . Corollary 2.7. [13, Theorem 3.3-Theorem 3.4]
Let H and K be Hermitian ma-trices such that mI ≤ H, K ≤ M I for some scalars m ≤ M , and let α ∈ [0 , .Then for all p > , λ k ( e (1 − α ) H + αK ) ≤ S ( e ( M − m ) p ) p λ k ( e pH ♯ α e pK ) p , k = 1 , , . . . , n. So, for every unitarily invariant norm k · k u k e (1 − α ) H + αK k u ≤ S ( e ( M − m ) p ) p k ( e pH ♯ α e pK ) p k u , and the right-hand side of these inequalities converges to the left-hand side as p ↓ .Proof. From mI ≤ H, K ≤ M I , we have e νm ≤ e νH , e νK ≤ e νM for every ν ≥ e m − M e H (cid:22) ols e K (cid:22) ols e M − m e H . Now the assertion isobtained by applying Theorem 2.5 and the fact that for every t > S ( t ) = S ( t ). (cid:3) reverse inequalities via kantorovich constant A well-known matrix version of the Kantorovich inequality [11] asserts that if A and U are two matrices such that 0 < mI ≤ A ≤ M I and
U U ∗ = I , then U A − U ∗ ≤ ( m + M ) mM ( U AU ∗ ) − . (3.1)Let w >
0. The generalized Kantorovich constant K ( w, α ) is defined by K ( w, α ) := w α − w ( α − w − (cid:0) α − α w α − w α − w (cid:1) α , (3.2)for any real number α ∈ R [6]. In fact, K ( Mm , −
1) = K ( Mm ,
2) is the constantoccurring in (3.1).Now as a result of following statement, we have another reverse Golden-Thompsontype inequality which refines corresponding inequality in [13].
Proposition 3.1. [8, Theorem 3]
Let H and K be Hermitian matrices such that e s e H (cid:22) ols e K (cid:22) ols e t e H for some scalars s ≤ t , and let α ∈ [0 , . Then λ k ( e (1 − α ) H + αK ) ≤ K ( e p ( t − s ) , α ) − p λ k ( e pH ♯ α e pK ) p , p > , (3.3) where K ( w, α ) is the generalized Kantorovich constant defined as (3.2) . Theorem 3.2.
Let H and K be Hermitian matrices such that mI ≤ K, H ≤ M I for some scalars m ≤ M and let α ∈ [0 , . Then for every p > λ k ( e (1 − α ) H + αK ) ≤ K ( e p ( M − m ) , α ) − p λ k ( e pH ♯ α e pK ) p , k = 1 , , . . . , n, M.B. GHAEMI, V. KALEIBARY AND S. FURUICHI and the right-hand side of this inequality converges to the left-hand side as p ↓ .In particular, λ k ( e H + K ) ≤ e M + e m e M e m λ k ( e H ♯e K ) , k = 1 , , . . . , n. Proof.
Since mI ≤ K, H ≤ M I implies e m − M e H (cid:22) ols e K (cid:22) ols e M − m e H , desiredinequalities are obtained by letting s = m − M and t = M − m in Proposition 3.1.For the convergence, we know that 2 w w + 1 ≤ K ( w, α ) ≤
1, for every α ∈ [0 , p > ≤ K ( w p , α ) − p ≤ ( 2 w p w p + 1 ) − p . A simple calculation shows thatlim p → − p Ln ( 2 w p w p + 1 ) = lim p → Ln ( w )( w P − w p + 1 = 0 , and hence lim p → ( 2 w p w p + 1 ) − p = 1. Now by using the sandwich condition andletting w = e M − m ) , we have lim p → K ( e p ( M − m ) , α ) − p = 1. (cid:3) Remark . Under the assumptions of Theorem 3.2, Seo in [13, Theorem 4.2]proved that k e (1 − α ) H + αK k u ≤ K ( e ( M − m ) , p ) − αp K ( e p ( M − m ) , α ) − p k ( e pH ♯ α e pK ) p k u , < p ≤ , and k e (1 − α ) H + αK k u ≤ K ( e p ( M − m ) , α ) − p k ( e pH ♯ α e pK ) p k u , p ≥ . But the inequality (3.3) shows that the sharper constant for all p > K ( e p ( M − m ) , α ) − p .Since for 0 < p ≤ K ( e ( M − m ) , p ) − αp ≥ K ( e p ( M − m ) , α ) − p ≤ K ( e ( M − m ) , p ) − αp K ( e p ( M − m ) , α ) − p . some related results It has been shown [7] that if f : [0 , ∞ ) −→ [0 , ∞ ) is operator monotone functionand 0 < mI ≤ A ≤ B ≤ M I ≤ I with h = Mm , then for all α ∈ [0 , f ( A ) ♯ α f ( B ) ≤ exp (cid:0) α (1 − α )(1 − h ) (cid:1) f ( A♯ α B ) , (4.1)This new ratio has been introduced by Furuichi and Minculete in [4], which isdifferent from Specht ratio and Kantorovich constant. By applying (4.4) for f ( t ) = t r , 0 < r ≤ Lemma 4.1.
Let A and B be positive definite matrices such that < mI ≤ A ≤ B ≤ M I ≤ I with h = M/m , and let α ∈ [0 , . Then A r ♯ α B r ≤ exp (cid:0) rα (1 − α )(1 − h ) (cid:1) ( A♯ α B ) r , < r ≤ , N REVERSES OF THE GOLDEN-THOMPSON TYPE INEQUALITIES 7
Lemma 4.2.
Let A and B be positive definite matrices such that < mI (cid:22) ols A (cid:22) ols B (cid:22) ols M I (cid:22) ols I with h = M/m , and let α ∈ [0 , . Then for all k = 1 , , . . . , n , λ k ( A♯ α B ) r ≤ exp (cid:0) α (1 − α )(1 − h r ) (cid:1) λ k ( A r ♯ α B r ) , r ≥ ,λ k ( A q ♯ α B q ) q ≤ exp (cid:0) p α (1 − α )(1 − h p ) (cid:1) λ k ( A p ♯ α B p ) p , < q ≤ p. (4.2) Theorem 4.3.
Let H and K be Hermitian matrices such that e m I (cid:22) ols e H (cid:22) ols e K (cid:22) ols e M I (cid:22) ols I for some scalars m ≤ M , and let α ∈ [0 , . Then for all p > and k = 1 , , . . . , nλ k ( e (1 − α ) H + αK ) ≤ exp (cid:0) p α (1 − α )(1 − e p ( M − m ) ) (cid:1) λ k ( e pH ♯ α e pK ) p , and so, for every unitarily invariant norm k · k u k e (1 − α ) H + αK k u ≤ exp (cid:0) p α (1 − α )(1 − e p ( M − m ) ) (cid:1) k ( e pH ♯ α e pK ) p k u . Proof.
The proof is similar to that of Theorem 2.5, by replacing A and B with e H and e K , and h = e M − m in the inequality (4.2). (cid:3) Remark . Under the different conditions, the different coefficients are not com-parable. But it is known that if we have a certain statement under the sandwichcondition 0 < sA ≤ B ≤ tA , then the same statement is also true under thecondition 0 < mI ≤ A, B ≤ M I and 0 < mI ≤ A ≤ B ≤ M I ≤ I . Hence, wecan compare the following special cases:(1) Comparison of the constants in Theorem 4.3 and in Theorem 3.2:Let e m I (cid:22) ols e H (cid:22) ols e K (cid:22) ols e M I (cid:22) ols I . Operator monotony of log( t )leads to mI ≤ H ≤ K ≤ M I ≤ I , and so mI ≤ H, K ≤ M I . Now byapplying Theorem 3.2 we have λ k ( e (1 − α ) H + αK ) ≤ K ( e p ( M − m ) , α ) − p λ k ( e pH ♯ α e pK ) p , p > . Also, by Theorem 4.3 λ k ( e (1 − α ) H + αK ) ≤ exp (cid:0) p α (1 − α )(1 − e p ( M − m ) ) (cid:1) λ k ( e pH ♯ α e pK ) p , p > . Letting h = e M − m ≥
1, the following numerical examples show that thereis no ordering between these inequalities.(i) Take α = , p = and h = 2, then we have K ( h p , α ) − p − exp (cid:0) p α (1 − α )(1 − h p ) (cid:1) ≃ − . . (ii) Take α = , p = and h = 8, then we have K ( h p , α ) − p − exp (cid:0) p α (1 − α )(1 − h p ) (cid:1) ≃ . . M.B. GHAEMI, V. KALEIBARY AND S. FURUICHI (2) Comparison of the constants in Lemma 4.1 and in Lemma 2.1:Let 0 < mI ≤ A ≤ B ≤ M I ≤ I . Then the following sandwich conditionis obtained m ≤ mM ≤ ≤ A − BA − ≤ Mm ≤ m . Now by letting s = 1 and t = Mm = h in Lemma 2.1, we get A r ♯ α B r ≤ S ( h ) r ( A♯ α B ) r , < r ≤ . (4.3)Also, by Lemma 4.1 A r ♯ α B r ≤ exp (cid:0) rα (1 − α )(1 − h ) (cid:1) ( A♯ α B ) r , < r ≤ . (4.4)It is shown in [4, Remark 2.4] that there is no ordering between coefficientsof (4.3) and (4.4). Therefore, we may conclude evaluation of Lemma 4.1and Lemma 2.1 are different. References
1. T. Ando and F. Hiai, Log-majorization and complementary Golden-Thompson type in-equalities, Linear Algebra Appl. 197/198 (1994) 113–131.2. R. Bhatia, Matrix Analysis, Grad. Texts in Math., vol. 169, Springer-Verlag, 1997.3. J.-C. Bourin and Y. Seo, Reverse inequality to Golden-Thompson type inequalities: com-parison of e A + B and e A e B , Linear Algebra Appl. (2007), 312–316.4. S. Furuichi and N. Minculete, Alternative reverse inequalities for Youngs inequality, J.Math. Inequal. 5 (2011) 595–600.5. T. Furuta, Operator inequalities associated with Holder-McCarthy and Kantorovich in-equalities, J. Inequal. Appl. 2 (1998) 137–148.6. T. Furuta, J. Mi´ci´c, J.E. Peˇcari´c and Y. Seo, Mond-Peˇcari´c method in operator inequalities,Monographs in Inequalities 1, Element, Zagreb, 2005.7. M.B. Ghaemi and V. Kaleibary, Some inequalities involving operator monotone functionsand operator means, Math. Inequal. Appl. 19 (2016) 757–764.8. M.B. Ghaemi and V. Kaleibary, Eigenvalue inequalities related to the Ando-Hiai inequality,Math. Inequal. Appl. 20 (2017) 217–223.9. F. Hiai and D. Petz, The Golden-Thompson trace inequality is complemented, Linear Al-gebra Appl. 181 (1993) 153–185.10. F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1980) 205–224.11. B. Mond and J.E. Peˇcari´c, A matrix version of the Ky Fan generalization of the Kantorovichinequality, Linear and Multilinear Algebra. 36 (1994) 217–221.12. M.P. Olson, The selfadjoint operators of a von Neumann algebra from a conditionallycomplete lattice, Proc. Amer. Math. Soc. 28 (1971) 537–544.13. Y. Seo, Reverses of the Golden–Thompson type inequalities due to Ando–Hiai–Petz, BanachJ. Math. Anal. 2 (2008) 140–149.14. Y. Seo, On a reverse of Ando-Hiai inequality, Banach J. Math. Anal. 4 (2010) 87–91.15. W. Specht, Zur Theorie der elementaren Mittel, Math. Z. 74 (1960) 91–98. School of Mathematics, Iran University of Science and Technology, Narmak,Tehran 16846-13114, Iran.
E-mail address : [email protected] School of Mathematics, Iran University of Science and Technology, Narmak,Tehran 16846-13114, Iran.
E-mail address : [email protected] N REVERSES OF THE GOLDEN-THOMPSON TYPE INEQUALITIES 9
Department of Information Science, College of Humanities and Sciences, Ni-hon University, 3-25-40, Sakurajyousui, Setagaya-ku, Tokyo, 156-8550, Japan.
E-mail address ::