ON JOINT CONVEXITY AND CONCAVITY OF SOME KNOWN TRACE FUNCTIONS

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1 ON JOINT CONVEXITY ND CONCVITY OF SOME KNOWN TRCE FUNCTIONS MOHMMD GHER GHEMI, NHID GHRKHNLU and YOEL JE CHO Communicated by Dan Timotin In this aer, we rovide a new and simle roof for joint convexity and concavity of some known trace functions due to ekjan. Indeed, we will do this by making use of the oerator monotone functions and ersective of convex functions. MS 200 Subject Classification: 55, Key words: trace functions, oerator monotone functions, joint convexity, joint concavity, ersective functions.. INTRODUCTION We are concerned with the joint concavity and convexity of some known trace functions. The concets and theorems on oerator monotone functions and ersective of convex functions lay an essential role in the method of roving our main results. Let H be a finite dimensional Hilbert sace, (H) be the sace of all bounded linear oerators from H to H, + (H) be the set of all ositive oerators in (H), s (H) be the set of all Hermitian oerators in (H) and ++ (H) be the set of all strictly ositive oerators in (H), resectively. real valued function f(, ) defined on (H) (H) is said to be jointly convex in (, ) if f(λ + ( λ) 2, λ + ( λ) 2 ) λf(, ) + ( λ)f( 2, 2 ) for all i, i (H), i 2 and λ [0,. f(, ) is said to be jointly concave if f(, ) is jointly convex in (, ). function f : (0, ) R is said to be oerator monotone if > 0 imlies that f() f(). We remark that for all, s (H) if 0. For more information on the oerator monotone functions and their relations with joint convexity and concavity, the reader is referred to [, 2, 7 and [2. Let f be a real valued function defined on a convex set C R n. The ersective function [0 associated to f is a function of two variables on the subset MTH. REPORTS 9(69), (207), 69 74

2 70 M.. Ghaemi, N. Gharakhanlu and Y.J. Cho 2 defined by L := { (t, s) s > 0, P f (t, s) = f ( ) } t C R n+ s ( ) t s. s If f(x) is convex on C, then P f (t, s) is jointly convex in (t, s). In [5, Effros introduced an oerator version of ersective functions for commuting ositive oerators L and R by ( ) L P f (L, R) = f R, R and roved the following theorem: Theorem. ([5, Theorem 2.2). If f(t) is oerator convex and [L, R = 0, then the ersective ( ) L P f (L, R) = f R, R is jointly convex. In [4, Carlen and Lieb roved the joint concavity of T r ( n) for all 0 < and, ++ (H). In 204, Hansen [8 gave a simle roof for joint convexity of T r( + ) when 2. In this aer, we give a new and simle roof for the joint concavity of T r( + ) when 0 <. We note that ekjan [3 has roved the joint concavity of T r( + ) by Estein s method, while we give easier roof by alying the ersective of convex functions and Effros s convexity theorem. In [9, Hiai and Petz considered the trace function I θ f (,, K) = T rk [ f(l R )R θ (K) for a ositive function f on (0, ), a non zero real arameter θ, an arbitrary oerator K on H and, ++ (H). They roved various roerties concerning joint concavity and convexity of the function If θ (,, K) in three variables (,, K) or in two variables (, ). Their main theorems clarified what conditions of f and θ are sufficient and necessary for joint convexity and concavity of If θ (,, K). For any 0 <, we rove the joint convexity of T r( + ) and T r( + ) by alying the joint convexity conditions of If θ (,, K).

3 3 On joint convexity and concavity 7 and Notice that the joint convexity of [ T r ( n ) [ T r ( n) was considered in [3, but in this aer, we introduce a simle and new roof. Throughout this aer, we suose that H = M n have the usual Hilbert sace structure with inner roduct, := T r. For all, ++ (H), we define L (X) := X and R (X) := X. They have the following roerties [: () L and R are commuting oerators since R [L (X) = X = L [R (X); (2) L and R are invertible, L = L and R = R ; (3) L = L and R = R for all R. 2. MIN RESULTS Now, we give our main results in this aer. Theorem 2. (ekjan). If 0<, then f(, ) = T r ( + ) is jointly concave for, ++ (H). Proof. Let g(t) = (t + ) defined for t > 0 and 0 <. y simle algebraic calculation, g(t) can be rewritten as follows: Since the function (t + ) t g(t) = (t + ) is oerator monotone ([, Corollary 4.3), we have g(t) = ( t + ) t = (t + ) is oerator monotone and oerator concave ([7, Corollary 2.6 and [2, Corollary 6). Now, we obtain the ersective of g(t) = (t + ) ( ) t P g (t, s) = g s = ( t + s ). s as follows:

4 72 M.. Ghaemi, N. Gharakhanlu and Y.J. Cho 4 So, for commuting oerators L and R, we have ( P g (L, R ) = L + R ) Since g(t) is oerator concave, P g (L, R ) is jointly concave by Theorem.. Thus we conclude that following maing [ (2.) (, ) T r (L + R ) (K )K is jointly concave in (, ) for arbitrary oerator K on H ([6, Theorem.). If we relace K with the identity oerator I in (2.), then the following maing [ ( (, ) T r L ) + R (I )I is jointly concave. ccording to the roof of ([8, Theorem 2.), we have [ (L + ) [ (2.2) R (I) = ( + ) and so, by a simle algebraic calculation, we obtain [ ( ) T r L [ ( ) (I )I =T r L (I) + R + R. [ ( =T r + ). Thus [ ( f(, ) = T r + ) is jointly concave in (, ). This comletes the roof. Corollary 2.2. If and 0, then T r ( + ) concave for any, ++ (H). is jointly Proof. T r ( + ) is jointly concave for any 0 < ([8, Theorem 2.). For any 0 <, T r ( + ) is jointly concave by Theorem 2.. So T r ( + ) is jointly concave for < 0 and this comletes the roof. Remark 2.3. We notice that f(t) = (t +) is oerator convex for t > 0 and 0 < ([2, Corollary V.2.6). The ersective function ( ) L P f (L, R ) = f R = (L + R ) R, 2 R is jointly convex from Theorem. which means that the joint convexity of g(, ) = T r ( + ) is not followed by alying the method which is

5 5 On joint convexity and concavity 73 used in Theorem 2.. similar argument is hold for T r ( + ) with f(t) = (t + ) (note that f(t) = (t + ) is oerator convex by ([2, Theorem 4)). Now, we are ready to give our new roofs for the joint convexity of trace functions T r ( + ) and T r ( + ). We do this according to [9 (Theorem 3.2 and Theorem 3.3) and so, first, we state a necessary art of them as a lemma for the convenience of the reader. Lemma 2.4. Let f is an oerator monotone function on (0, ) and θ (0,. Then T rk [ f(l R )R θ (K) is jointly convex for all, ++ (H) and an arbitrary oerator K on H. Theorem 2.5 (ekjan). Let 0 < and, ++ (H). T r ( + ) and T r ( + ) are jointly convex in (, ). Then Proof. t first, we rove the joint convexity of T r ( + ). Let f(t) = (t + ) for any t > 0 and 0 <. Since f(t) is an oerator monotone function, the following trace function (2.3) T rk [ f(l R )R [ (K) = T rk (L + R ) (K) is jointly convex in (, ) according to Lemma 2.4 with θ =. Now, if we substitute K with the identity oerator I in (2.3), then it follows that [ T ri (L + R ) (I) = T ri ( L + ) R (I) = T r ( L + ) (2.4) R (I) is jointly convex in (, ). ccording to (2.2), (2.4) and a simle calculation, we have T r ( L + ) R (I) = T r ( + ), which means that T r ( + ) joint convexity of T r ( + ) is jointly convex in (, ). Similarly, the can be roved. It is sufficient to relace the oerator monotone function f(t) = (t +) for any t > 0 and 0 < in Lemma 2.4 with θ =. This comletes the roof.

6 74 M.. Ghaemi, N. Gharakhanlu and Y.J. Cho 6 cknowledgments. Yeol Je Cho was suorted by asic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (204R ). REFERENCES [ T. ndo, Concavity of certain mas on ositive definite matrices and alications to Hadamard roduct. Linear lgebra l. 26 (979), [2 R. hatia, Matrix nalysis. Grad. Texts in Math. Sringer, erlin, 997. [3 T.N. ekjan, On joint convexity of trace functions. Linear lgebra l. 390 (2004), [4 E.. Carlen and E.H. Lieb, Minkowski tye trace inequality and strong subadditivity of quantum entroy. In: Differential Oerators and Sectral Theory. mer. Math. Soc. Trans. 89 (999), [5 E.G. Effros, matrix convexity aroach to some celebrated quantum inequalities. Proc. Natl. cad. Sci. US 06 (2009), [6 F. Hansen, Extensions of Lieb s concavity theorem. J. Stat. Phys. 24 (2006), [7 F. Hansen and G.K. Pedersen, Jensen s inequality for oerators and Löwner s theorem. Math. nn. 258 (982), [8 F. Hansen, Trace functions with alications in quantum hysics. J. Stat. Phys. 54 (204), [9 F. Hiai and D. Petz, Convexity of quasi-entroy functions: Lieb s and ndo s convexity theorems revisited. J. Math. Phys. 54 (203), 2. [0 J. Hiriart-Urruty and C. Lemarchal, Fundamental of Convex nalysis. Grundlehren Text Ed, Sringer, erlin, 200. [. Jencova and M.. Ruskai, unified treatment of relative entroy and related trace functions with conditions for equality. Rev. Math. Phys. 22 (200), [2 M.K. Kwong, Some results on matrix monotone functions. Linear lgebra l. 8 (989), Received 4 July 205 Iran University of Science and Technology, School of Mathematics, Narmak, Tehran , Iran mghaemi@iust.ac.ir Iran University of Science and Technology, School of Mathematics, Narmak, Tehran , Iran gharakhanlu nahid@mathde.iust.ac.ir Gyeongsang National University, Deartment of Mathematics Education and the RINS, Chinju , Korea and King bdulaziz University, Deartment of Mathematics, Jeddah 2589, Saudi rabia yjcho@gnu.ac.kr