SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES

Transcription

1 Kragujevac Journal of Mathematics Volume 411) 017), Pages SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES SILVESTRU SEVER DRAGOMIR 1, Abstract. Some new trace ineualities for oerators in Hilbert saces are rovided. The sueradditivity and monotonicity of some associated functionals are investigated and alications for ower series of such oerators are given. Some trace ineualities for matrices are also derived. Examles for the oerator exonential and other similar functions are resented as well. 1. Introduction Let H,, ) be a comlex Hilbert sace and {e i } i I an orthonormal basis of H. We say that A B H) is a Hilbert-Schmidt oerator if 1.1) Ae i <. i I It is well know that, if {e i } i I and {f j } j J are orthonormal bases for H and A B H) then 1.) Ae i = Af j = A f j i I j I j I showing that the definition 1.1) is indeendent of the orthonormal basis and A is a Hilbert-Schmidt oerator iff A is a Hilbert-Schmidt oerator. Key words and hrases. Trace class oerators, Hilbert-Schmidt oerators, trace, Schwarz ineuality, trace ineualities for matrices, ower series of oerators. 010 Mathematics Subject Classification. Primary: 47A63. Secondary: 47A99. Received: May 18, 016. Acceted: July 4,

2 34 S. S. DRAGOMIR Let B H) the set of Hilbert-Schmidt oerators in B H). For A B H) we define ) 1 A := Ae i, i I for {e i } i I an orthonormal basis of H. This definition does not deend on the choice of the orthonormal basis. Using the triangle ineuality in l I), one checks that B H) is a vector sace and that is a norm on B H), which is usually called in the literature as the Hilbert-Schmidt norm. Denote the modulus of an oerator A B H) by A := A A) 1. Because A x = Ax for all x H, A is Hilbert-Schmidt iff A is Hilbert- Schmidt and A = A. From 1.) we have that if A B H), then A B H) and A = A. The following theorem collects some of the most imortant roerties of Hilbert- Schmidt oerators. Theorem 1.1. We have: i) B H), ) is a Hilbert sace with inner roduct A, B := i I Ae i, Be i = i I B Ae i, e i and the definition does not deend on the choice of the orthonormal basis {e i } i I ; ii) We have the ineualities A A for any A B H) and AT, T A T A for any A B H) and T B H); iii) B H) is an oerator ideal in B H), i.e. B H) B H) B H) B H) ; iv) B fin H), the sace of oerators of finite rank, is a dense subsace of B H); v) B H) K H), where K H) denotes the algebra of comact oerators on H. If {e i } i I an orthonormal basis of H, we say that A B H) is trace class if A 1 := i I A e i, e i <. The definition of A 1 does not deend on the choice of the orthonormal basis {e i } i I. We denote by B 1 H) the set of trace class oerators in B H). The following roosition holds. Proosition 1.1. If A B H), then the following are euivalent: i) A B 1 H);

3 SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES 35 ii) A 1 B H); iii) A or A ) is the roduct of two elements of B H). The following roerties are also well known. Theorem 1.. With the above notations: i) We have A 1 = A 1 and A A 1 for any A B 1 H); ii) B 1 H) is an oerator ideal in B H), i.e. iii) We have iv) We have B H) B 1 H) B H) B 1 H) ; B H) B H) = B 1 H) ; A 1 = su { A, B B B H), B 1 } ; v) B 1 H), 1 ) is a Banach sace; vi) We have the following isometric isomorhisms B 1 H) = K H) and B 1 H) = B H), where K H) is the dual sace of K H) and B 1 H) is the dual sace of B 1 H). We define the trace of a trace class oerator A B 1 H) to be 1.3) tr A) := i I Ae i, e i, where {e i } i I an orthonormal basis of H. Note that this coincides with the usual definition of the trace if H is finite-dimensional. We observe that the series 1.3) converges absolutely and it is indeendent from the choice of basis. The following result collects some roerties of the trace. Theorem 1.3. We have i) If A B 1 H) then A B 1 H) and tr A ) = tr A); ii) If A B 1 H) and T B H), then AT, T A B 1 H) and tr AT ) = tr T A) and tr AT ) A 1 T ; iii) tr ) is a bounded linear functional on B 1 H) with tr = 1; iv) If A, B B H) then AB, BA B 1 H) and tr AB) = tr BA); v) B fin H) is a dense subsace of B 1 H).

4 36 S. S. DRAGOMIR Utilizing the trace notation we obviously have that A, B = tr B A) = tr AB ) and A = tr A A) = tr A ) for any A, B B H). Now, for the finite dimensional case, it is well known that the trace functional is submultilicative, that is, for ositive semidefinite matrices A and B in M n C), 0 trab) tr A) tr B). Therefore 0 tra k ) [tr A)] k, where k is any ositive integer. In 000, Yang [17] roved a matrix trace ineuality 1.4) tr [ AB) k] tr A) k tr B) k, where A and B are ositive semidefinite matrices over C of the same order n and k is any ositive integer. For related works the reader can refer to [3, 4, 11, 18], which are continuations of the work of Bellman []. If H,, ) is a searable infinite-dimensional Hilbert sace then the ineuality 1.4) is also valid for any ositive oerators A, B B 1 H). This result was obtained by L. Liu in 007, see [9]. In 001, Yang et al. [16] imroved 1.4) as follows 1.5) tr [AB) m ] [ tr A m) tr B m)] 1, where A and B are ositive semidefinite matrices over C of the same order and m is any ositive integer. In [13] the authors have roved many trace ineualities for sums and roducts of matrices. For instance, if A and B are ositive semidefinite matrices in M n C) then 1.6) tr [ AB) k] min { A k tr B k), B k tr A k)} for any ositive integer k. Also, if A, B M n C) then for r 1 and, > 1 with 1/ + 1/ = 1 we have the following Young tye ineuality [ A ) r ] 1.7) tr AB r ) tr + B. Ando [1] roved a very strong form of Young s ineuality it was shown that if A and B are in M n C), then there is a unitary matrix U such that 1 AB U A + 1 ) B U, where, > 1 with 1/ + 1/ = 1, which immediately gives the trace ineuality tr AB ) 1 tr A ) + 1 tr B ). This ineuality can also be obtained from 1.7) by taking r = 1.

5 SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES 37 Another Hölder tye ineuality has been roved by Manjegani in [10] and can be stated as follows: 1.8) trab) [tra )] 1 [trb )] 1, where, > 1 with 1/ + 1/ = 1 and A and B are ositive semidefinite matrices. For the theory of trace functionals and their alications the reader is referred to [14]. For other trace ineualities see [3, 6 8, 1, 15]. In this aer, motivated by the above results, some new ineualities for Hilbert- Schmidt oerators in B H) are rovided. The sueradditivity and monotonicity of some associated functionals are investigated and alications for ower series of such oerators are given. Some trace ineualities for matrices are also derived. Examles for the oerator exonential and other similar functions are resented as well.. Some Trace Ineualities via Hermitian Forms Let P a selfadjoint oerator with P 0. For A B H) and {e i } i I an orthonormal basis of H we have A,P := tr A P A) = i I P Ae i, Ae i P i I Ae i = P A, which shows that,,p defined by A, B,P := tr B P A) = i I P Ae i, Be i = i I B P Ae i, e i is a nonnegative Hermitian form on B H), i.e.,,p satisfies the roerties: h) A, A,P 0 for any A B H); hh),,p is linear in the first variable; hhh) B, A,P = A, B,P for any A, B B H). Using the roerties of the trace we also have the following reresentations A,P := tr P A ) = tr A P ) and A, B,P := tr P AB ) = tr AB P ) = tr B P A) for any A, B B H). We start with the following result. Theorem.1. Let P a selfadjoint oerator with P 0, i.e. P x, x 0 for any x H. i) For any A, B B H) we have.1) tr P AB ) tr P A ) tr P B ) and.) [ tr P A ) + Re tr P AB ) + tr P B )] 1

6 38 S. S. DRAGOMIR [ tr P A )] 1 + [ tr P B )] 1 ; ii) For any A, B, C B H) we have.3) tr P AB ) tr P C ) tr P AC ) tr P CB ) [ tr P A ) tr P C ) tr P AC ) ] [ tr P B ) tr P C ) tr P BC ) ],.4) and.5) tr P AB ) tr P C ) tr P AB ) tr P C ) tr P AC ) tr P CB ) + tr P AC ) tr P CB ) [ tr P A )] 1 [ tr P B )] 1 tr P C ) tr P AC ) tr P CB ) 1 [ [tr P A )] 1 [ tr P B )] 1 ] + tr P AB ) tr P C ). Proof. i) Making use of the Schwarz ineuality for the nonnegative hermitian form,,p we have A, B,P A, A,P B, B,P for any A, B B H) and the ineuality.1) is roved. We observe that,p is a seminorm on B H) and by the triangle ineuality we have A + B,P A,P + B,P for any A, B B H) and the ineuality.) is roved. ii) Let C B H), C 0. Define the maing [, ],P,C : B H) B H) C by [A, B],P,C := A, B,P C,P A, C,P C, B,P. Observe that [, ],P,C is a nonnegative Hermitian form on B H) and by Schwarz ineuality we have A, B,P C,P A, C,P C, B.6),P [ A ] [,P A, C,P ] C,P B,P B, C,P C,P for any A, B B H), which roves.3). The case C = 0 is obvious. Utilizing the elementary ineuality for real numbers m, n,, m n ) ) m n),

7 SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES 39 we can easily see that [ A,P C,P A, C,P ] [ B,P C,P B, C,P ].7) ) A,P B,P C,P A, B, C,P C,P for any A, B, C B H). Since, by Schwarz s ineuality we have and A,P C,P B,P C,P then by multilying these ineualities we have A, C,P B, C,P, A,P B,P C,P A, C,P B, C,P, for any A, B, C B H). Utilizing the ineualities.6) and.7) and taking the suare root we get A, B,P C,P A, C,P C, B.8),P A,P B,P C A, B,,P C,P C,P for any A, B, C B H), which roves the second ineuality in.4). The first ineuality is obvious by the modulus roerties. By the triangle ineuality for modulus we also have A, A, C,P C, B,P B,P C.9),P A, B,P C,P A, C,P C, B,P for any A, B, C B H). On making use of.8) and.9) we have A, A, C,P C, B,P B,P C,P A,P B,P C,P A, C,P B, C,P, which is euivalent to the desired ineuality.5). Remark.1. By the triangle ineuality for the hermitian form [, ],P,C : B H) B H) C, [A, B],P,C := A, B,P C,P A, C,P C, B,P

8 40 S. S. DRAGOMIR we have A + B,P C,P A + B, C,P ) 1 A ),P A, C,P 1 ) C,P + B B,,P C,P 1 C,P, which can be written as [ tr P A + B) ] tr P C ) tr [P A + B) C ] ) 1.10) for any A, B, C B H). tr P A ) tr P C ) tr P AC ) ) 1 + tr P B ) tr P C ) tr P BC ) ) 1 Remark.. If we take B = λc in.10), then we get [.11) 0 tr P A + λc) ] tr P C ) tr [P A + λc) C ] tr P A ) tr P C ) tr C P A), for any λ C and A, C B H). Therefore, we have the bound {.1) tr [P A + λc) ] tr P C ) tr [P A + λc) C ] } su λ C = tr P A ) tr P C ) tr P AC ). We also have the ineualities [.13) 0 tr P A ± C) ] tr P C ) tr [P A ± C) C ] for any A, C B H). tr P A ) tr P C ) tr P AC ), Remark.3. We observe that, by relacing A with A, B with B etc. above, we can get the dual ineualities, like, for instance.14) tr P A C) tr P C B) 1 [ [tr P A )] 1 [ tr P B )] 1 ] + tr P A B) tr P C ), that holds for any A, B, C B H). Since tr P A C) = tr P A C) = tr [P A C) ] = tr C AP ) = tr P C A), tr P C B) = tr P B C)

9 SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES 41 and tr P A B) = tr P B A) then the ineuality.14) can be also written as.15) tr P C A) tr P B C) 1 [ [tr P A )] 1 [ tr P B )] 1 ] + tr P B A) tr P C ), that holds for any A, B, C B H). If we take in.15) B = A then we get the following ineuality.16) tr P C A) tr P AC) 1 [ [tr P A )] 1 [ tr P A )] 1 + ) tr P A ] tr P C ), for any A, B, C B H). If A is a normal oerator, i.e. A = A then we have from.16) that tr P C A) tr P AC) 1 [ tr P A ) + ) tr P A ] tr P C ), In articular, if C is selfadjoint and C B H), then.17) tr P AC) 1 [ tr P A ) + ) tr P A ] tr P C ), for any A B H) a normal oerator. We notice that.17) is a trace oerator version of de Bruijn ineuality obtained in 1960 in [5], which gives the following refinement of the Cauchy-Bunyakovsky-Schwarz ineuality [.18) a i z i 1 ] a i z i + zi, i=1 i=1 rovided that a i are real numbers while z i are comlex for each i {1,..., n}. We notice that, if P B 1 H), P 0 and A, B B H), then i=1 i=1 A, B,P := tr P AB ) = tr AB P ) = tr B P A) is a nonnegative Hermitian form on B H) and all the ineualities above will hold for A, B, C B H). The details are left to the reader. 3. Some Functional Proerties We consider now the convex cone B + H) of nonnegative oerators on the comlex Hilbert sace H and, for A, B B H) define the functional σ A,B : B + H) [0, ) by 3.1) σ A,B P ) := [ tr P A )] 1 [ tr P B )] 1 tr P A B) 0). The following theorem collects some fundamental roerties of this functional.

10 4 S. S. DRAGOMIR Theorem 3.1. Let A, B B H). i) For any P, Q B + H) we have 3.) σ A,B P + Q) σ A,B P ) + σ A,B Q) 0), namely, σ A,B is a sueradditive functional on B + H) ; ii) For any P, Q B + H) with P Q we have 3.3) σ A,B P ) σ A,B Q) 0), namely, σ A,B is a monotonic nondecreasing functional on B + H) ; iii) If P, Q B + H) and there exist the constants M > m > 0 such that MQ P mq then 3.4) Mσ A,B Q) σ A,B P ) mσ A,B Q) 0). Proof. i) Let P, Q B + H). On utilizing the elementary ineuality a + b ) 1 c + d ) 1 ac + bd, a, b, c, d 0 and the triangle ineuality for the modulus, we have σ A,B P + Q) = [ tr P + Q) A )] 1 [ tr P + Q) B )] 1 tr P + Q) A B) = [ tr P A + Q A )] 1 [ tr P B + Q B )] 1 tr P A B + QA B) = [ tr P A ) + tr Q A )] 1 [ tr P B ) + tr Q B )] 1 tr P A B) + tr QA B) [ tr P A )] 1 [ tr P B )] 1 + [ tr Q A )] 1 [ tr Q B )] 1 tr P A B) tr QA B) =σ A,B P ) + σ A,B Q) and the ineuality 3.) is roved. ii) Let P, Q B + H) with P Q. Utilizing the sueradditivity roerty we have σ A,B P ) = σ A,B P Q) + Q) σ A,B P Q) + σ A,B Q) σ A,B Q) and the ineuality 3.3) is obtained. iii) From the monotonicity roerty we have σ A,B P ) σ A,B mq) = mσ A,B Q) and a similar ineuality for M, which rove the desired result 3.4). Corollary 3.1. Let A, B B H) and P B H) such that there exist the constants M > m > 0 with M1 H P m1 H. Then we have [tr M A )] 1 [ tr B )] 1 ) 3.5) tr A B)

11 SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES 43 [ tr P A )] 1 [ tr P B )] 1 tr P A B) [tr m A )] 1 [ tr B )] 1 ) tr A B). Let P = V with V B H). If A, B B H) then σ A,B V ) = [ tr V A )] 1 [ tr V B )] 1 tr V A B ) = [tr V V A A)] 1 [tr V V B B)] 1 tr V V A B) = [tr V A AV )] 1 [tr V B BV )] 1 tr V A BV ) = [tr AV ) AV )] 1 [tr BV ) BV )] 1 tr AV ) BV ) = [ tr AV )] 1 [ tr BV )] 1 tr AV ) BV ). On utilizing the roerty 3.) for P = V, Q = U with V, U B H), then we have for any A, B B H) the following trace ineuality [ tr AV + AU )] 1 [ tr BV + BU )] 1 3.6) tr AV ) BV + AU ) BU ) [ tr AV )] 1 [ tr BV )] 1 tr AV ) BV ) + [ tr AU )] 1 [ tr BU )] 1 tr AU ) BU ) 0). Also, if V U with V, U B H), then we have for any A, B B H) that [ tr AV )] 1 [ tr BV )] 1 3.7) tr AV ) BV ) [ tr AU )] 1 [ tr BU )] 1 tr AU ) BU ) 0). If U B H) is invertible, then 1 x Ux U x, for any x H, U 1 which imlies that 1 U 1 1 H U U 1 H. By making use of 3.5) we have the following trace ineuality 3.8) for any A, B B H). U [ tr A )] 1 [ tr B )] 1 tr A B) [ tr AU )] 1 [ tr BU )] 1 tr AU ) BU ) 1 [tr A )] 1 [ U 1 tr B )] 1 ) tr A B) )

12 44 S. S. DRAGOMIR Similar results may be stated for P B 1 H), P 0 and A, B B H). The details are omitted. 4. Ineualities for Seuences of Oerators For n, define the Cartesian roducts B n) H) := B H) B H), B n) H) := B H) B H) and B n) + H) := B + H) B + H), where B + H) denotes the convex cone of nonnegative selfadjoint oerators on H, i.e. P B + H) if P x, x 0 for any x H. Proosition 4.1. Let P = P 1,..., P n ) B n) + H) and A = A 1,..., A n ), B = B 1,..., B n ) B n) H) and z = z 1,..., z n ) C n with n. Then ) ) ) 4.1) tr z k P k A kb k tr z k P k A k tr z k P k B k. Proof. Using the roerties of modulus and the ineuality.1) we have ) tr z k P k A kb k = z k tr P k A kb k ) z k tr P k A kb k ) z k [ tr P k A k )] 1 [ tr P k B k )] 1. Utilizing the weighted discrete Cauchy-Bunyakovsky-Schwarz ineuality we also have z k [ tr P k A k )] 1 [ tr P k B k )] 1 ) 1 [tr z k Pk A k )] 1 ) z k = z k tr 1 P k A k )) z k tr 1 P k B k )) ) 1 [tr Pk B k )] 1 ) )) 1 )) 1 = tr z k P k A k tr z k P k B k, which is euivalent to the desired result 4.1).

13 SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES 45 We consider the functional for n-tules of nonnegative oerators as follows: [ )] 1 [ )] 1 4.) σ A,B P) := tr P k A k tr P k B k ) tr P k A kb k. Utilizing a similar argument to the one in Theorem 3.1 we can state the following. Proosition 4.. Let A = A 1,..., A n ), B = B 1,..., B n ) B n) H). i) For any P, Q B n) + H) we have 4.3) σ A,B P + Q) σ A,B P) + σ A,B Q) 0), namely, σ A,B is a sueradditive functional on B n) + H); ii) For any P, Q B n) + H) with P Q, namely P k Q k for all k {1,..., n} we have 4.4) σ A,B P) σ A,B Q) 0), namely, σ A,B is a monotonic nondecreasing functional on B n) + H); iii) If P, Q B n) + H) and there exist the constants M > m > 0 such that MQ P mq then 4.5) Mσ A,B Q) σ A,B P) mσ A,B Q) 0). If P = 1 1 H,..., n 1 H ) with k 0, k {1,..., n} then the functional of nonnegative weights = 1,..., n ) defined by [ )] 1 [ )] 1 ) σ A,B ) := tr k A k tr k B k tr k A kb k. has the same roerties as in 4.3) 4.5). Moreover, we have the simle bounds: [ )] 1 [ )] 1 ) max { k} tr A k tr B k 4.6) k {1,...,n} tr A kb k [ )] 1 [ )] 1 ) tr k A k tr k B k tr k A kb k [ )] 1 [ )] 1 ) min { k} tr A k tr B k k {1,...,n} tr A kb k.

14 46 S. S. DRAGOMIR Denote by: 5. Ineualities for Power Series of Oerators D0, R) = and consider the functions and { {z C : z < R}, if R <, C, if R =, λ fλ) : D0, R) C, fλ) := λ f a λ) : D0, R) C, f a λ) := α n λ n α n λ n. As some natural examles that are useful for alications, we can oint out that, if 1) n 1 f λ) = n λn = ln, 1 + λ n=1 λ D 0, 1) ; 1) n g λ) = n)! λn = cos λ, λ C; h λ) = l λ) = 1) n n + 1)! λn+1 = sin λ, λ C; 1) n λ n = 1, λ D 0, 1) ; 1 + λ then the corresonding functions constructed by the use of the absolute values of the coefficients are 5.1) 1 1 f a λ) = n λn = ln, 1 λ n=1 λ D 0, 1) ; 1 g a λ) = n)! λn = cosh λ, λ C; h a λ) = l a λ) = 1 n + 1)! λn+1 = sinh λ, λ C; λ n = 1, λ D 0, 1). 1 λ Other imortant examles of functions as ower series reresentations with nonnegative coefficients are: 1 5.) ex λ) = n! λn, λ C;

15 SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES 47 ) λ ln = 1 λ sin 1 λ) = tanh 1 λ) = F 1 α, β, γ, λ) = 1 n 1 λn 1, λ D 0, 1) ; Γ ) n + 1 λ n+1, λ D 0, 1) ; π n + 1) n! n=1 n=1 where Γ is Gamma function. 1 n 1 λn 1, λ D 0, 1) ; Γ n + α) Γ n + β) Γ γ) n!γ α) Γ β) Γ n + γ) λn, α, β, γ > 0, λ D 0, 1) ; Proosition 5.1. Let fλ) := α nλ n be a ower series with comlex coefficients and convergent on the oen disk D 0, R), R > 0. If H,, ) is a searable infinite-dimensional Hilbert sace and A, B B 1 H) are ositive oerators with tr A), tr B) < R 1, then 5.3) tr f AB)) f a tr A tr B) f a tr A) ) f a tr B) ). Proof. By the ineuality 1.4) for the ositive oerators A, B B 1 H) we have [ ] tr α k AB) k = α k tr [ AB) k] 5.4) α k [ ] tr AB) k = α k tr [ AB) k] α k tr A) k tr B) k = α k tr A tr B) k. Utilizing the weighted Cauchy-Bunyakovsky-Schwarz ineuality for sums we have ) 1 ) 1 5.5) α k tr A) k tr B) k α k tr A) k α k tr B) k. Then by 5.4) and 5.5) we have [ ] [ ] 5.6) tr α k AB) k α k tr A tr B) k for n 1. α k [ tr A) ] k n α k [ tr B) ] k

16 48 S. S. DRAGOMIR Since 0 tr A), tr B) < R 1, the numerical series α k tr A tr B) k, α k [ tr A) ] k and α k [ tr B) ] k are convergent. Also, since 0 trab) tr A) tr B) < R, the oerator series α kab) k is convergent in B 1 H). Letting n in 5.6) and utilizing the continuity roerty of tr ) on B 1 H) we get the desired result 5.3). Examle 5.1. a) If we take in 5.3) fλ) = 1 ± λ) 1, λ < 1 then we get the ineuality tr 1 H ± AB) 1) 1 tr A) ) 1 1 tr B) ) 1 for any A, B B 1 H) ositive oerators with tr A), tr B) < 1. b) If we take in 5.3) fλ) = ln 1 ± λ) 1, λ < 1, then we get the ineuality 5.7) tr ln 1 H ± AB) 1) ln 1 tr A) ) 1 ln 1 tr B) ) 1, for any A, B B 1 H) ositive oerators with tr A), tr B) < 1. We have the following result as well. Theorem 5.1. Let fλ) := α nλ n be a ower series with comlex coefficients and convergent on the oen disk D 0, R), R > 0. If A, B B H) are normal oerators with A B = BA and tr A ), tr B ) < R, then we have the ineuality 5.8) tr f A B)) tr f a A )) tr f a B )). Proof. From the ineuality 4.1) we have ) 5.9) tr α k A ) k B k tr α k ) A k tr α k ) B k. Since A, B are normal oerators, then we have A k = A k and B k = B k for any k 0. Also, since A B = BA, then we also have A ) k B k = A B) k for any k 0. Due to the fact that A, B B H) and tr A ), tr B ) < R, it follows that tr A B) R and the oerator series α k A B) k, α k A k and α k B k are convergent in the Banach sace B 1 H). Taking the limit over n in 5.9) and using the continuity of the tr ) on B 1 H) we deduce the desired result 5.8).

17 SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES 49 Examle 5.. a) If we take in 5.8) fλ) = 1 ± λ) 1, λ < 1 then we get the ineuality tr 1H ± A B) 1) 1 tr A ) ) 1 1 tr B ) ) 1 for any A, B B H) normal oerators with A B = BA and tr A ), tr B ) < 1. b) If we take in 5.8) fλ) = ex λ), λ C then we get the ineuality 5.10) tr ex A B)) tr ex A )) tr ex B )) for any A, B B H) normal oerators with A B = BA. Theorem 5.. Let f z) := j=0 jz j and g z) := j=0 jz j be two ower series with nonnegative coefficients and convergent on the oen disk D 0, R), R > 0. If T and V are two normal and commuting oerators from B H) with tr T ), tr V ) < R, then [ tr f T ) + g T ))] 1 [ tr f V ) + g V ))] ) tr f T V ) + g T V )) [ tr f T ))] 1 [ tr f V ))] 1 tr f T V )) + [ tr g T ))] 1 [ tr g V ))] 1 tr g T V )) 0). Moreover, if j j for any j N, then, with the above assumtions on T and V, we have [ tr f T ))] 1 [ tr f V ))] 1 5.1) tr f T V )) [ tr g T ))] 1 [ tr g V ))] 1 tr g T V )) 0). Proof. Utilizing the sueradditivity roerty of the functional σ A,B ) above as a function of weights and the fact that T and V are two normal and commuting oerators we can state that 5.13) for any n 1. [ )] 1 [ tr k + k ) T k tr k + k ) V k )] 1 ) tr k + k ) T V ) k [ )] 1 [ )] 1 ) tr k T k tr k V k tr k T V ) k [ )] 1 [ )] 1 ) + tr k T k tr k V k tr k T V ) k

18 50 S. S. DRAGOMIR Since all the series whose artial sums are involved in 5.13) are convergent in B 1 H), by letting n in 5.13) we get 5.11). The ineuality 5.1) follows by the monotonicity roerty of σ A,B ) and the details are omitted. Examle 5.3. Now, observe that if we take 1 f λ) = sinh λ = n + 1)! λn+1 and then g λ) = cosh λ = f λ) + g λ) = ex λ = 1 n)! λn 1 n! λn for any λ C. If T and V are two normal and commuting oerators from B H), then by.11) we have [ tr ex T ))] 1 [ tr ex V ))] 1 tr ex T V )) [ tr sinh T ))] 1 [ tr sinh V ))] 1 tr sinh T V )) + [ tr cosh T ))] 1 [ tr cosh V ))] 1 tr cosh T V )) 0). 1 Now, consider the series 1 λ λ D 0, 1) and define n = 1, n 0, 0 = 0, n = 1 n = λn, λ D 0, 1) and ln 1 1 λ = 1 n=1 n λn,, n 1, then we observe that for any n 0 we have n n. If T and V are two normal and commuting oerators from B H) with tr T ), tr V ) < 1, then by.1) we have [ 1H tr T ) )] 1 1 [ 1H tr V ) )] 1 1 tr 1H T V ) 1) [ tr ln 1 H T ) )] 1 1 [ tr ln 1 H V ) )] 1 1 tr ln 1H T V ) 1) 0). 6. Ineualities for Matrices We have the following result for matrices. Proosition 6.1. Let fλ) := α nλ n be a ower series with comlex coefficients and convergent on the oen disk D 0, R), R > 0. If A and B are ositive semidefinite matrices in M n C) with tr A ), tr B ) < R, then we have the ineuality 6.1) tr [fab)] tr [ f a A )] tr [ f a B )].

19 SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES 51 If tr A), tr B) < R, then also 6.) tr [fab)] min {tr f a A B)), tr f a B A))}. Proof. We observe that 1.5) holds for m = 0 with euality. By utilizing the generalized triangle ineuality for the modulus and the ineuality 1.5) we have [ m ] tr α n AB) n = α n tr [AB) n ] m 6.3) α n tr [AB) n ] = α n tr [AB) n ] α n [ tr A n)] 1 [ tr B n)] 1, for any m 1. Alying the weighted Cauchy-Bunyakowsky-Schwarz discrete ineuality we also have α n [ tr A n)] 1 [ tr B n)] 1 6.4) m [tr α n )] ) 1 ) A n 1 m [tr α n )] ) 1 ) B n 1 m = α n [ tr 1 A n)]) m α n [ tr 1 B n)]) [ m )] 1 [ m )] 1 = tr α n A n tr α n B n for any m 1. Therefore, by 6.3) and 6.4) we get [ m ] 6.5) tr m ) m ) α n AB) n tr α n A n tr α n B n for any m 1. Since tr A ), tr B ) < R, then tr AB) tr A ) tr B ) < R and the series α n AB) n, α n A n and α n B n are convergent in M n C). Taking the limit over m in 6.5) and utilizing the continuity roerty of tr ) on M n C) we get 6.1). The ineuality 6.) follows from 1.6) in a similar way and the details are omitted.

20 5 S. S. DRAGOMIR Examle 6.1. a) If we take fλ) = 1 ± λ) 1, λ < 1 then we get the ineuality [ tr In ± AB) 1] [ In tr A ) ] [ 1 In tr B ) ] 1 for any A and B ositive semidefinite matrices in M n C) with tr A ), tr B ) < 1. Here I n is the identity matrix in M n C). We also have the ineuality [ tr In ± AB) 1] { min tr In A B) 1), tr I n B A) 1)} for any A and B ositive semidefinite matrices in M n C) with tr A), tr B) < 1. b) If we take fλ) = ex λ, then we have the ineualities tr [exab)]) tr [ ex A )] tr [ ex B )] and tr [exab)] min {tr ex A B)), tr ex B A))} for any A and B ositive semidefinite matrices in M n C). Proosition 6.. Let fλ) := α nλ n be a ower series with comlex coefficients and convergent on the oen disk D 0, R), R > 0. If A and B are matrices in M n C) with tr A ), tr B ) < R, where, > 1 with 1/ + 1/ = 1, then A 6.6) tr f AB )) tr [f a + B )] tr [ 1 f a A ) + 1 f a B ) Proof. The ineuality 1.7) holds with euality for r = 0. By utilizing the generalized triangle ineuality for the modulus and the ineuality 1.7) we have 6.7) tr m ) α n AB n = α n tr AB n ) α n tr AB n ) = = tr [ A α n tr [ m + B α n tr AB n ) A α n + B for any m 1 and, > 1 with 1/ + 1/ = 1. It is known that if f : [0, ) R is a convex function, then tr f ) is convex on the cone M n + C) of ositive semidefinite matrices in M n C). Therefore, for n 1 we have [ A tr + B ) n ] ) n ] ) n ] 1 tr A n ) + 1 tr B n ) ].

21 SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES 53 where, > 1 with 1/ + 1/ = 1. The ineuality reduces to euality if n = 0. Then we have 6.8) [ A α n tr ) n ] + B = tr [ 1 for any m 1 and, > 1 with 1/ + 1/ = 1. From 6.7) and 6.8) we get m ) [ tr m 6.9) α n AB r tr tr [ 1 [ 1 α n tr A n ) + 1 ] tr B n ) α n A n + 1 A ) ] n α n + B α n A n + 1 ] α n B n ] α n B n for any m 1 and, > 1 with 1/ + 1/ = 1. Since tr A ), tr B ) < R, then all the series whose artial sums are involved in 6.9) are convergent, then by letting m in 6.9) we deduce the desired ineuality 6.6). Examle 6.. a) If we take fλ) = 1 ± λ) 1, λ < 1 then we get the ineualities [ tr In ± AB ) 1) A )] ) 1 tr I n + B [ 1 tr I n A ) ] I n B ) 1, where A and B are matrices in M n C) with tr A ), tr B ) < 1 and, > 1 with 1/ + 1/ = 1. b) If we take fλ) = ex λ, then we have the ineualities [ A tr ex AB )) tr ex tr + B )] [ 1 ex A ) + 1 ex B ) where A and B are matrices in M n C) and, > 1 with 1/ + 1/ = 1. Finally, we can state the following result. Proosition 6.3. Let fλ) := α nλ n be a ower series with comlex coefficients and convergent on the oen disk D 0, R), R > 0. If A and B are commuting ositive ],

22 54 S. S. DRAGOMIR semidefinite matrices in M n C) with tr A ), tr B ) < R, where, > 1 with 1/ + 1/ = 1, then we have the ineuality tr f AB)) [trf a A ))] 1 [trfa B ))] 1. Proof. Since A and B are commuting ositive semidefinite matrices in M n C), then by 1.8) we have for any natural number n including n = 0 that 6.10) trab) n ) = tra n B n ) [tra n )] 1 [trb n )] 1, where, > 1 with = 1. By 6.10) and the weighted Hölder discrete ineuality we have m ) tr α n AB) n = α n tra n B n ) m α n tra n B n ) α n [tra n )] 1 [trb n )] 1 m ) 1 ) α n [tra n )] 1 m = α n tra n ) = tr α n A n ) ) 1 ) 1 m m α n trb n ) tr α n B n ) ) 1 ) α n [trb n )] 1 where, > 1 with 1/ + 1/ = 1. The roof follows now in a similar way with the ones from above and the details are omitted. Examle 6.3. a) If we take fλ) = 1 ± λ) 1, λ < 1 then we get the ineuality ) 1 ) 1 tr In ± AB) 1) [ trin A ) 1 ) ] 1 [ tri n B ) 1 ) ] 1, for any A and B commuting ositive semidefinite matrices in M n C) with tr A ), tr B ) < 1, where, > 1 with 1/ + 1/ = 1. b) If we take fλ) = ex λ, then we have the ineuality tr ex AB)) [trex A ))] 1 [trex B ))] 1, for any A and B commuting ositive semidefinite matrices in M n C) and, > 1 with 1/ + 1/ = 1.,

23 SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES 55 References [1] T. Ando, Matrix young ineualities, Oer. Theory Adv. Al ), [] R. Bellman, Some ineualities for ositive definite matrices, in: E. F. Beckenbach Ed.), Proceedings of the nd International Conference on General Ineualities, Birkhäuser, Basel, 1980, [3] D. Chang, A matrix trace ineuality for roducts of Hermitian matrices, J. Math. Anal. Al ), [4] I. D. Coo, On matrix trace ineualities and related toics for roducts of hermitian matrix, J. Math. Anal. Al ), [5] N. G. de Bruijn, Problem 1, Wisk. Ogaven ), [6] E.-V. Belmega, M. Jungers and S. Lasaulce, A generalization of a trace ineuality for ositive definite matrices, Aust. J. Math. Anal. Al ), 1 5. [7] S. Furuichi and M. Lin, Refinements of the trace ineuality of Belmega, Lasaulce and Debbah, Aust. J. Math. Anal. Al ), 1 4. [8] H. D. Lee, On some matrix ineualities, Korean J. Math ), [9] L. Liu, A trace class oerator ineuality, J. Math. Anal. Al ), [10] S. Manjegani, Hölder and young ineualities for the trace of oerators, Positivity ), [11] H. Neudecker, A matrix trace ineuality, J. Math. Anal. Al ), [1] K. Shebrawi and H. Albadawi, Oerator norm ineualities of Minkowski tye, J. Ineual. Pure Al. Math ), [13] K. Shebrawi and H. Albadawi, Trace ineualities for matrices, Bull. Aust. Math. Soc ), [14] B. Simon, Trace Ideals and their Alications, Cambridge University Press, Cambridge, [15] Z. Ulukök and R. Türkmen, On some matrix trace ineualities, J. Ineual. Al ), 1 8. [16] X. M. Yang, X. Q. Yang and K. L. Teo, A matrix trace ineuality, J. Math. Anal. Al ), [17] X. Yang, A matrix trace ineuality, J. Math. Anal. Al ), [18] Y. Yang, A matrix trace ineuality, J. Math. Anal. Al ), Mathematics, School of Engineering & Science Victoria University PO Box 1448, Melbourne City, MC 8001 Australia address: sever.dragomir@vu.edu.au DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences School of Comutational & Alied Mathematics University of the Witwatersrand Private Bag 3, Johannesburg 050 South Africa