In this paper, we introduce the notion of multivariate generalized perspectives and verify the necessary and sufficient conditions for operator convexity (concavity) of this notion. We also establish the crossing of the multivariate generalized perspective of regular operator mappings under completely positive linear maps and partial traces

Ismail Nikoufar

Moosa Shamohammadi

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MATHEMATICAL COMMUNICATIONS 1Math. Commun. 23(2018), 1–8Generalized perspectives of functions of several variablesIsmail Nikoufar∗and Moosa ShamohammadiDepartment of Mathematics, Payame Noor University, P.O. Box 19 395-3 697 Tehran,IranReceived July 5, 2016; accepted February 18, 2017Abstract. In this paper, we introduce the notion of multivariate generalized perspectivesand verify the necessary and sufficient conditions for operator convexity (resp. concavity)of this notion. We also establish the crossing of the multivariate generalized perspective ofregular operator mappings under completely positive linear maps and partial traces.AMS subject classifications: Primary 81P45; Secondary 94A17, 26B25Key words: multivariate generalized perspective, multivariate perspective, operator con-vexity1. Introduction and preliminariesEffros [4] considered the case where each pair in the argument of the perspectivefunction consists of commuting operators and proved in this way that the perspec-tive of an operator convex function is operator convex as a function. A fully non-commutative perspective of the one variable function f defined in [3] by settingPf(A,B) = B1/2f(B−1/2AB−1/2)B1/2and the generalized perspective of two variables (associated with f and h) definedbyPf∆h(A,B) = h(B)1/2f(h(B)−1/2Ah(B)−1/2)h(B)1/2,where A is a self-adjoint operator and B is a strictly positive operator on a Hilbertspace H with spectra in the closed interval I containing 0. Note that the identityPf∆h(A,B) = Pf (A, h(B)) expresses the generalized perspective in terms of thenon-commutative perspective. The main results of [4] are generalized in [3] for thenon-commutative case and the necessary and sufficient conditions for joint convex-ity (resp. concavity) of the perspective and generalized perspective functions areprovided. As an application of these results, Nikoufar et al. [10] gave the simplestproof of Lieb concavity and Ando convexity theorem (see also [11, 12, 1]).Hansen [6] introduced the notion of regular operator mappings of several variablesgeneralizing the notion of the spectral function of Davis for functions of one variable.Then, he generalized the notion of the perspective of a regular mapping of severalvariables and defined the geometric mean for any number of operator variables.∗Corresponding author. Email addresses: nikoufar@pnu.ac.ir (I. Nikoufar), m sh648@yahoo.com(M. Shamohammadi)http://www.mathos.hr/mc c©2018 Department of Mathematics, University of Osijek2 I. Nikoufar and M. ShamohammadiZhang [13] established operator concavity (resp. convexity) of some functions of twoor three variables by using the perspectives of the regular operator mappings of oneor several variables and specified operator concavity (resp. convexity) of the Fréchetdifferential mapping associated with some functions.Let B(H) and B(H)sa be the C∗-algebra of all bounded linear operators and theC∗-subalgebra of all self–adjoint bounded linear operators on a Hilbert space H,respectively. The notion of a regular mapping generalizes the notion of the spectralfunction of Davis for functions of one variable, the notion of the regular matrixmapping of two variables [8], and the notion of the regular operator mapping oftwo variables [5, Definition 2.1]. In [6, Definition 2.1], Hansen defined this notionas follows. Let F : Dn −→ B(H) be a mapping of n variables defined in a convexdomain Dn ⊆ B(H)sa×. . .×B(H)sa. Then, F is regular if the domain Dn is invariantunder unitary transformations of H andF (u∗x1u, . . . , u∗xnu) = u∗F (x1, . . . , xn)ufor every (x1, . . . , xn) ∈ Dn and every unitary u on H. For mutually orthogonal pro-jections p and q acting on H and arbitrary n-tuples (x1, . . . , xn) and (y1, . . . , yn) ofoperators inDn such that the compressed tuples (px1p, . . . ,pxnp) and (qy1q, . . . ,qynq)are in the domain Dn, the n-tuple of diagonal block matrices (px1p+qy1q, . . . , pxnp+qynq) is also in the domain Dn andF (px1p+ qy1q, . . . , pxnp+ qynq) = pF (px1p, . . . , pxnp)p+ qF (qy1q, . . . , qynq)q.Denote by D+n the positive convex domain of strictly positive operators A1, . . . , Anacting on a Hilbert space H. Let F : D+n −→ B(H) be a regular mapping. Theperspective mapping PF of n+ 1 variables is the mappingPF (A1, . . . , An, B) = B1/2F (B−1/2A1B−1/2, . . . , B−1/2AnB−1/2)B1/2defined in the domain D+n+1 for strictly positive operators A1, . . . , An and B actingon a Hilbert space H.In Section 2, we define multivariate generalized perspectives and prove a sub-homogeneous form of Jensen’s inequality for regular operator mappings of severalvariables. This leads us under some conditions to demonstrate the multivariategeneralized perspectives are operator convex (resp. concave). In Section 3, wedeclare the crossing of the multivariate generalized perspective of regular operatormappings through completely positive linear maps and partial traces.2. Multivariate generalized perspectivesIn this section, we define the multivariate generalized perspective of regular operatormappings of several variables. We confirm a sub-homogeneous form of Jensen’sinequality for regular operator mappings of several variables. We apply this to proveoperator convexity (resp. concavity) of the multivariate generalized perspectives.For other works on operator functions of several variables see [13] and referencestherein.Generalized perspectives of functions of several variables 3We know that the perspective of an operator convex function is operator con-vex as a function of two variables [3, Theorem 2.2]. Hansen proved a similar re-sult for a regular mapping of n operator variables [6]. Since PF (A1, . . . , An, 1) =F (A1, . . . , An), we confirm that the converse of this result is also true.Theorem 1. Let F : D+n −→ B(H) be a regular mapping. The perspective PF isoperator convex if and only if F is operator convex.Corollary 1. Let F : D+n −→ B(H) be a regular mapping. If F is operator convex(resp. concave), then PF is jointly subadditive (resp. superadditive).Proof. We observe that PF is positively homogenous in the sense thatPF (αA1, . . . , αAn, αB) = αPF (A1, . . . , An, B)for α > 0. Let {Ai1, Ai2, . . . , Ain} and {B1, B2, . . . , Bn} be two sequences of strictlypositive operators on a Hilbert space H for i = 1, .., n. According to Theorem 1 andoperator convexity of F , we havePF(n∑i=1Ai1, . . . ,n∑i=1Ain,n∑i=1Bi)= PF((n+ 1)n∑i=11n+ 1Ai1, . . . , (n+ 1)n∑i=11n+ 1Ain, (n+ 1)n∑i=11n+ 1Bi)= (n+ 1)PF(n∑i=11n+ 1Ai1, . . . ,n∑i=11n+ 1Ain,n∑i=11n+ 1Bi)≤ (n+ 1)n∑i=11n+ 1PF(Ai1, . . . , Ain, Bi)=n∑i=1PF(Ai1, . . . , Ain, Bi).After this, throughout the paper we assume that h is a real valued and continuousfunction defined on [0,∞). We say that h is strictly positive if h(A) > 0 for everyA > 0.Definition 1. Let F : D+n −→ B(H) be a regular mapping. The generalized perspec-tive mapping PF∆h (associated with F and h) is the mapping defined in the domainD+n+1 by settingPF∆h(A1, . . . , An, B)= h(B)1/2F (h(B)−1/2A1h(B)−1/2, . . . , h(B)−1/2Anh(B)−1/2)h(B)1/2for strictly positive operators A1, . . . , An and B acting on H.4 I. Nikoufar and M. ShamohammadiIt should be noted that the identity PF∆h(A1, . . . ,An, B) = PF (A1, . . . ,An, h(B))declares the generalized perspective in terms of Hansen’s perspective.In [6] Hansen proved an affine form of Jensen’s inequality for the regular operatormappings. For our purpose we prove the following sub-homogeneous form of Jensen’sinequality for the regular operator mappings. A sub-homogeneous form of Jensen’sinequality for the functions of one variable is proved in [7]. The proof of the followingresult is similar to that of [6, Theorem 2.2] (see also [9, Theorem 2.2]) and we omitit. Here, we state the result for operators T1 and T2 acting on a Hilbert space Hwith T ∗1 T1 + T∗2 T2 ≤ 1. In [6, Theorem 2.2] the result is proved for the case whereT ∗1 T1 + T∗2 T2 = 1. We need to apply the following theorem to our main resultTheorem 3.Theorem 2. Let F : Dn −→ B(H)sa be a convex regular mapping. If F (0, . . . , 0) ≤ 0and T1, T2 are operators acting on H with T∗1 T1 + T∗2 T2 ≤ 1, thenF (T ∗1A1T1 + T∗2A′1T2, . . . , T∗1AnT1 + T∗2A′nT2)≤ T ∗1F (A1, . . . , An)T1 + T∗2 F (A′1, . . . , A′n)T2(1)for n-tuples (A1, . . . , An) and (A′1, . . . , A′n) in Dn.In the following theorem, we extend [4, Theorem 3.2] and [3, Theorem 2.5, Corol-lary 2.6] for the regular mappings of several variables.Theorem 3. Suppose that h is a strictly positive function on (0,∞) and F : D+n −→B(H) is a regular mapping.(i) If F is operator convex and h is operator concave with F (0, . . . , 0) ≤ 0, thenthe generalized perspective mapping PF∆h is operator convex.(ii) If the generalized perspective mapping PF∆h is operator convex (resp. concave),then F is operator convex (resp. concave).(iii) If F (0, . . . , 0) > 0 and the generalized perspective mapping PF∆h is operatorconvex (resp. concave), then h is operator convex (resp. concave).(iv) If F (0, . . . , 0) < 0 and the generalized perspective mapping PF∆h is operatorconvex (resp. concave), then h is operator concave (resp. convex).Proof. (i): Let (A1, . . . , An, B1) and (A′1, . . . , A′n, B2) be in D+n+1 and 0 ≤ c ≤ 1.Put B = cB1 + (1− c)B2. DefineT1 = c1/2h(B1)1/2h(B)−1/2,T2 = (1− c)1/2h(B2)1/2h(B)−1/2.So, T ∗1 T1+T∗2 T2 ≤ 1, by operator concavity of h. From operator convexity of F andGeneralized perspectives of functions of several variables 5Theorem 2, we havePF∆h(cA1 + (1 − c)A′1, . . . , cAn + (1− c)A′n, cB1 + (1 − c)B2)= h(B)1/2F(h(B)−1/2(cA1 + (1− c)A′1)h(B)−1/2, . . . ,h(B)−1/2(cAn + (1− c)A′n)h(B)−1/2)h(B)1/2= h(B)1/2F(T ∗1 h(B1)−1/2A1h(B1)−1/2T1 + T∗2 h(B2)−1/2A′1h(B2)−1/2T2, . . . ,T ∗1 h(B1)−1/2Anh(B1)−1/2T1 + T∗2 h(B2)−1/2A′nh(B2)−1/2T2)h(B)1/2≤ h(B)1/2(T ∗1F (h(B1)−1/2A1h(B1)−1/2, . . . , h(B1)−1/2Anh(B1)−1/2)T1+ T ∗2 F (h(B2)−1/2A′1h(B2)−1/2, . . . , h(B2)−1/2A′nh(B2)−1/2)T2)h(B)1/2= ch(B1)1/2F (h(B1)−1/2A1h(B1)−1/2, . . . , h(B1)−1/2Anh(B1)−1/2)h(B1)1/2+ (1 − c)h(B2)1/2F (h(B2)−1/2A′1h(B2)−1/2, . . . ,h(B2)−1/2A′nh(B2)−1/2)h(B2)1/2= cPF∆h(A1, . . . , An, B1) + (1− c)PF∆h(A′1, . . . , A′n, B2).(ii): Note that F (A1, . . . , An) =1h(1)PF∆h(h(1)A1, . . . , h(1)An, 1). We haveF (cA1 + (1 − c)A′1, . . . , cAn + (1− c)A′n)=1h(1)PF∆h(ch(1)A1 + (1− c)h(1)A′1, . . . , ch(1)An + (1 − c)h(1)A′n), 1)≤1h(1)(cPF∆h(h(1)A1, . . . , h(1)An, 1) + (1− c)PF∆h(h(1)A′1, . . . , h(1)A′n, 1))= cF (A1, . . . , An) + (1− c)F (A′1, . . . , A′n).(iii): It is clear that h(B) = 1F (0,...,0)PF∆h(0, . . . , 0, B) and so thath(cB1 + (1− c)B2) =1F (0, . . . , 0)PF∆h(0, . . . , 0, cB1 + (1− c)B2)≤cF (0, . . . ,0)PF∆h(0, . . . , 0, B1)+1− cF (0, . . . , 0)PF∆h(0, . . . , 0, B2)= ch(B1) + (1 − c)h(B2).(iv) The proof is similar to that of (iii).Remark 1. As a simple consequence of Theorem 3 (i), if F and h are operatorconcave with F (0, . . . , 0) ≥ 0, then the generalized perspective mapping PF∆h isoperator concave.Corollary 2. Suppose that h is a strictly positive function defined on [0,∞) andF : D+n −→ B(H) is a regular mapping. Let h be operator concave and positivelyhomogenous.6 I. Nikoufar and M. Shamohammadi(i) If F is operator convex with F (0, . . . , 0) ≤ 0, then the generalized perspectivemapping PF∆h is jointly subadditive.(ii) If F is operator concave with F (0, . . . , 0) ≥ 0, then the generalized perspectivemapping PF∆h is jointly superadditive.Proof. (i): Let {Ai1, Ai2, . . . , Ain} and {B1, B2, . . . , Bn} be two sequences of strictlypositive operators on a Hilbert space H for i = 1, .., n. Since h is positively homoge-nous, we observe that PF∆h is jointly homogenous. Regarding Theorem 3 (i), Pf∆his jointly convex and soPF∆h(n∑i=1Ai1, . . . ,n∑i=1Ain,n∑i=1Bi)= PF∆h((n+ 1)n∑i=11n+ 1Ai1, . . . , (n+ 1)n∑i=11n+ 1Ain, (n+ 1)n∑i=11n+ 1Bi)= (n+ 1)Pf∆h(n∑i=11n+ 1Ai1, . . . ,n∑i=11n+ 1Ain,n∑i=11n+ 1Bi)≤ (n+ 1)n∑i=11n+ 1PF∆h(Ai1, . . . , Ain, Bi)=n∑i=1Pf∆h(Ai1, . . . , Ain, Bi).(ii): It follows from Remark 1.Corollary 3. Suppose that h is a strictly positive function defined on [0,∞) andF : D+n −→ B(H) is a regular mapping.(i) If the generalized perspective mapping PF∆h is operator convex (resp. concave)and F is positively homogenous, then F is jointly subadditive (resp. superad-ditive).(ii) If F (0, . . . , 0) > 0, the generalized perspective mapping PF∆h is operator convex(resp. concave), and h is positively homogenous, then h is subadditive (resp.superadditive).(iii) If F (0, . . . , 0) < 0, the generalized perspective mapping PF∆h is operator convex(resp. concave), and h is positively homogenous, then h is superadditive (resp.subadditive).3. Generalized perspectives and completely positive linearmapsIn this section, we study the filtering of the multivariate generalized perspectiveof a regular operator mapping through completely positive linear maps and partialtraces.Generalized perspectives of functions of several variables 7Theorem 4. Let Φ : B(H) → B(K) be a completely positive linear mapping betweenoperators on Hilbert spaces of finite dimensions, and let F : D+n → B(K) be a convexregular mapping. Then,PF∆h(Φ(A1), . . . ,Φ(An),Φ(B)) ≤ Φ(PF∆h(A1, . . . , An, B)) (2)for operators (A1, . . . , An, B) in D+n+1, where PF∆h is the generalized perspectivemapping (associated to F and h) and Φ◦h = h◦Φ on the strictly positive operators.Proof. Due to [9, Theorem 3.1], we getPF∆h(Φ(A1), . . . ,Φ(An),Φ(B)) = PF (Φ(A1), . . . ,Φ(An), h(Φ(B)))= PF (Φ(A1), . . . ,Φ(An),Φ(h(B))≤ Φ(PF (A1, . . . , An, h(B)))= Φ(PF∆h(A1, . . . , An, B)).Remark 2. Under the hypotheses of Theorem 4, if F is a concave regular mapping,then the reverse inequality is valid in (2).We justify the other conditions under which the reverse inequality in (2) is valid.Definition 2. Let F : D+n → B(K) be a mapping. We say that F is n-monotone if itis monotone on its nth variable, in the sense that if An ≤ Bn, then F (A1, . . . , An) ≤F (B1, . . . , Bn) for (A1, . . . , An), (B1, . . . , Bn) in D+n .Theorem 5. Let Φ : B(H) → B(K) be a completely positive linear mapping betweenoperators on Hilbert spaces of finite dimensions, and let F : D+n → B(K) be a concaveregular mapping. If PF is (n+ 1)-monotone and h is concave with h(0) = 0, thenPF∆h(Φ(A1), . . . ,Φ(An),Φ(B)) ≥ Φ(PF∆h(A1, . . . , An, B)). (3)Proof. Applying [2, Corollary], we get h(Φ(B)) ≥ Φ(h(B)). Since F is concave,the reverse inequality holds in [9, Theorem 3.1], so that by the (n+1)-monotonicityproperty of PF we concludePF∆h(Φ(A1), . . . ,Φ(An),Φ(B)) = PF (Φ(A1), . . . ,Φ(An), h(Φ(B)))≥ PF (Φ(A1), . . . ,Φ(An),Φ(h(B))≥ Φ(PF (A1, . . . , An, h(B)))= Φ(PF∆h(A1, . . . , An, B)).Remark 3. Under the hypotheses of Theorem 5, if F is convex and regular and his convex, then the reverse inequality is valid in (3).We know that for a bipartite system H = H1⊗H2, Tr2A is the partial trace of Aon H1. The function h is the partial trace preserving on H1 if h(Tr2A) = Tr2h(A).The fact that the partial trace is completely positive leads to the following corollaries.8 I. Nikoufar and M. ShamohammadiCorollary 4. Consider a bipartite system H = H1 ⊗ H2 of Hilbert spaces H1 andH2 of finite dimensions. If F : D+n → B(K) is a convex regular mapping and h ispartial trace preserving on H1, thenPF∆h(Tr2A1, . . . , T r2An, T r2B) ≤ Tr2PF∆h(A1, . . . , An, B)for operators (A1, . . . , An, B) in D+n+1.Corollary 5. Consider a bipartite system H = H1 ⊗ H2 of Hilbert spaces H1 andH2 of finite dimensions and a concave regular mapping F : D+n → B(K). If PF is(n+ 1)-monotone and h is concave with h(0) = 0, thenPF∆h(Tr2A1, . . . , T r2An, T r2B) ≥ Tr2(PF∆h(A1, . . . , An, B))for operators (A1, . . . , An, B) in D+n+1.AcknowledgementThe first author was supported by a grant from Payame Noor University, Iran.References[1] Z.Chen, Z.Ma, I. Nikoufar, S.-M. Fei, Sharp continuity bounds for entropy andconditional entropy, Sci. China Phys. Mech. Astron. 60(2017), 1–5.[2] C.Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc.8(1957), 42–44.[3] A.Ebadian, I. Nikoufar, M.Eshagi Gordji, Perspectives of matrix convex func-tions, Proc. Natl. Acad. Sci. 108(2011), 7313–7314.[4] E.G.Effros, A matrix convexity approach to some celebrated quantum inequalities,Proc. Natl. Acad. Sci. USA 106(2009), 1006–1008.[5] E.G.Effros, F. Hansen, Non-commutative perspectives, Ann. Funct. Anal. 5(2014),74–79.[6] F.Hansen, Regular operator mappings and multivariate geometric means, Linear Al-gebra Appl. 461(2014), 123–138.[7] F.Hansen, G.Pedersen, Jensen’s inequality for operators and Lowner’s theorem,Math. Ann. 258(1982), 229-241.[8] F.Hansen,Means and concave products of positive semi-definite matrices, Math. Ann.264(1983), 119-128.[9] F.Hansen, Perspectives and completely positive maps, Ann. Funct. Anal. 8(2017),168-176.[10] I. Nikoufar, A.Ebadian, M. Eshagi Gordji, The simplest proof of Lieb concavitytheorem, Adv. Math. 248(2013), 531–533.[11] I. Nikoufar, On operator inequalities of some relative operator entropies, Adv. Math.259(2014), 376–383.[12] I. Nikoufar, A perspective approach for characterization of Lieb concavity theorem,Demonstratio Math. 49(2016), 463–469.[13] Z. Zhang, Some operator convex functions of several variables, Linear Algebra Appl.463(2014), 1–9.

Generalized perspectives of functions of several variables

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