Gselmann, Eszter

Aequationes Mathematicae

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Jordan triple mappings on positive definitematricesEszter GselmannAbstract. The main aim of this paper is to characterize the determinantfunction on the set of positive definite n× n matrices with entries fromF.Mathematics Subject Classification (2010). Primary 39B42; Secondary15A30.Keywords. Jordan triple mapping, determinant, characterization, posi-tive definite matrix.1. Introduction and preliminariesThe characterization of the determinant function is one of the oldest questionin the field of linear algebra. The standard approach to the study of deter-minants is to prove the existence, for any n-dimensional vector space X overa field F, of a nonzero alternating multilinear form d : Xn → F and to definethe determinant function det : L (X)→ F by the requirementdet(T ) · d (x1, . . . , xn) = d (Tx1, . . . , Txn) (T ∈ L (X))for all (x1, . . . , xn) ∈ Xn. These conditions determine d up to a multiplicativeconstant and hence determine the function det. See e.g. Fearnley-Sander [5].It would be however desirable to characterize the determinant functionthrough a certain identity. For instance, it is well-known that the determinantfunction fulfills the identityχ(A ·B) = χ(A) · χ(B) (1)and also the equationψ(A ·B ·A) = ψ(A) · ψ(B) · ψ(A) (2)This research has been supported by the Hungarian Scientific Research Fund (OTKA)Grant NK 814 02 and by the ’Lendu¨let’ Program (LP2012-46/2012) of the HungarianAcademy of Sciences.2 Eszter Gselmannfor all A,B ∈ L (X). Mappings ψ : L (X)→ F satisfying the previous equa-tion are termed to be Jordan triple mappings.The main aim of this paper is to characterize the determinant functionon the set of positive definite n × n matrices with entries from F. In thissetting, equation (1) is obviously inappropriate, since the ordinary product oftwo positive definite matrices will not be in general a positive definite matrix.However, the Jordan triple product of two such matrix will be again positive.Thus, in what follows we will investigate equation (2), or more precisely, itsadditive counterpart, namely the following equationφ(A ·B ·A) = 2φ(A) + φ(B).The notion of Jordan triple mappings were firstly introduced in Yam-aguti [12]. Further, the characterization of such type of mappings began inthe 1960’s, see for example Ga´spa´r [6], Hitotumatu [7], Kurepa [9]. Concern-ing Jordan triple mappings, the results are rather recent, see Doboviˇsek [4],Bobecka–Weso lowski [3] and Molna´r [11].In what follows, we will shortly review the known results in this area.In [3] differentiable Jordan triple mappings φ : Pn(R)→]0,+∞[ were deter-mined.Theorem 1 (Bobecka–Weso lowski). Let φ : Pn(R) →]0,+∞[ be a differen-tiable mapping so thatφ(√Y X√Y)= φ(X)φ(Y )is fulfilled for all X,Y ∈Pn(R). Then there exists κ ∈ R so thatφ(X) = det(X)κ (X ∈Pn(R)) .Later, using a theorem of Gleason, it was showed that differentiabilitycan be weakened and continuous Jordan triple mappings φ : Pn(C)→ R werecompletely described in [11], assuming that n ≥ 3. This method however doesnot work in case P2(C).Theorem 2 (Molna´r). Let n ≥ 3 be arbitrarily fixed and φ : Pn(C)→ R be acontinuous functional. Assume that for all X,Y ∈Pn(C) functional equationφ(Y XY ) = φ(Y )φ(X)φ(Y )is fulfilled. Then either φ is identically zero or there exists κ ∈ R such thatφ(X) = det(X)κ (X ∈Pn(R)) .Without any regularity assumption, Jordan triple mappings were foundout in [4], though not for mappings φ : Pn(F) → R but for φ : Mn(F) → R.Our main aim is to investigate Jordan triple mappings in their natural setting.More precisely, we would like to solve functional equationφ(Y XY ) = 2φ(Y ) + φ(X) (X,Y ∈Pn(F))for the unknown function φ : Pn(F)→ R without any regularity suppositionand irrespective of the value n.Jordan triple mappings 3Firstly we will fix the notation and the terminology that will be usedsubsequently. Concerning the theory of functional equations, we refer to thetwo basic monographs Acze´l [1] and Kuczma [8].Definition 3. Let A ⊂]0,+∞[ be a nonempty set andL ={(x, y) ∈]0,+∞[2 |x, y, x · y ∈ A} .A function ` : A→ R is termed to be a logarithmic function on A if`(xy) = `(x) + `(y)is fulfilled for any (x, y) ∈ L . In case A =]0,+∞[, then the function ` willsimply be called logarithmic.In what follows R, C will stand for the set of the real, respectivelycomplex numbers.Definition 4. Let F ∈ {R,C} and n ∈ N be fixed, thenMn(F) = {X |X is an n× n matrix with entries from F}andPn(F) = {X ∈Mn(F) |X is positive definite} .Henceforward, concerning matrix analysis, the notions and results of thebasic monograph Bhatia [2] will be utilized. The following easy statementswill be used permanently.Lemma 1. Let n ∈ N and F ∈ {R,C}. Then(i) X ∈ Pn(F) if and only if X is Hermitian and all its principal minorsare positive.(ii) X ∈Pn(F) if and only if all its eigenvalues are positive.(iii) X ∈Pn(F) if and only if there exists Y ∈Pn(F) such thatX = Y 2,such Y is uniquely determined. We will write Y =√X and call it thesquare root of X.2. The 2× 2 caseAt first we prove the following. Henceforth we always assume F ∈ {R,C}.Theorem 5. Let φ : P2(F)→ R be a mapping for whichφ(Y XY ) = φ(X) + 2φ(Y )holds for all X,Y ∈ P2(F). Then, there exists a logarithmic function l :]0,+∞[→ R such thatφ(X) = l (det(X)) (X ∈P2(F)) .4 Eszter GselmannProof. Let X,Y ∈P2(F) be diagonal matrices, then obviously X ·Y = Y ·X,thereforeφ(X · Y 2) = φ(X) + 2φ(Y ).If we replace Y by√Y , we obtain thatφ(X · Y ) = φ(X) + φ(Y ) (∗)is valid for all diagonal matrices X,Y ∈P2(F). Let X,Y ∈P2(F) be diago-nal matrices, say,X =(λ1 00 λ2)and Y =(µ1 00 µ2),where λ1, λ2, µ1, µ2 are positive real numbers. ThenX · Y =(λ1µ1 00 λ2µ2)Define the function ϕ : ]0,+∞[2→ R byϕ(λ1, λ2) = φ(λ1 00 λ2)(λ1, λ2 ∈]0,+∞[) .Then due to identity (∗), we haveϕ(λ1µ1, λ2µ2) = ϕ(λ1, λ2) + ϕ(µ1, µ2) (λ1, λ2, µ1, µ2 ∈]0,+∞[) .With the choice λ2 = µ1 = 1 and µ2 = λ2 this equation yields thatϕ(λ1, λ2) = ϕ(λ1, 1) + ϕ(1, λ2)holds for all λ1, λ2 ∈]0,+∞[. On the other hand (∗) with λ2 = µ2 = 1 impliesthatϕ(λ1µ1, 1) = ϕ(λ1, 1) + ϕ(µ1, 1) (λ1, µ1 ∈]0,+∞[) ,further, identity (∗) with the choice λ1 = µ1 = 1 yields thatϕ(1, λ2µ2) = ϕ(1, λ2) + ϕ(1, µ2) (λ2, µ2 ∈]0,+∞[) .All in all, this means that there exist logarithmic functions l1, l2 : ]0,+∞[→ Rsuch thatl1(λ) = ϕ(λ, 1) and l2(λ) = ϕ(1, λ) (λ ∈]0,+∞[)is fulfilled. Therefore for all λ, µ ∈]0,+∞[φ(λ 00 µ)= ϕ(λ, µ) = l1(λ) + l2(µ)is satisfied.Let now X ∈P2(F)X =(a bb d)be arbitrary. ThenY ·X · Y =(1 b√a√db√a√d1),Jordan triple mappings 5whereY =(1√a00 1√d).Let U = {x ∈ F : ‖x‖ < 1} and define the function f : U → R throughf(x) = φ(1 xx 1).LetX =(1 xx 1)(x ∈ U)be arbitrarily fixed. Thenφ(X−1) = −φ(X) = −f(x).On the other hand,φ(X−1) = φ− 1x2 − 1 xx2 − 1xx2 − 1 −1x2 − 1= φ1√1− x2 001√1− x2 · ( 1 −x−x 1)·1√1− x2 001√1− x2= −l(√1− x2)+ f(−x)− l(√1− x2).All in all, this implies thatf(x) + f(−x) = 2l(√1− x2)(x ∈ U) .From this it follows however thatf(x) =12l(1− x2) + g(x) (x ∈ U) ,where g : U → R is an odd function.Due to the definition of the mapping f , we also havef(x)+2f(y) = φ(1 xx 1)+2φ(1 yy 1)= φ((1 yy 1)·(1 xx 1)·(1 yy 1))= φ(y2 + 2x y + 1 x y2 + 2 y + xx y2 + 2 y + x y2 + 2x y + 1)6 Eszter Gselmannholds for all x, y ∈ U . On the other hand, for all x, y ∈ U ,φ(y2 + 2x y + 1 x y2 + 2 y + xx y2 + 2 y + x y2 + 2x y + 1)= φ((√y2 + 2x y + 1 00√y2 + 2x y + 1)··(1 x y2+2 y+xy2+2 x y+1x y2+2 y+xy2+2 x y+1 1)·(√y2 + 2x y + 1 00√y2 + 2x y + 1))= 2{l1(√y2 + 2x y + 1)+ l2(√y2 + 2x y + 1)}+f(x y2 + 2 y + xy2 + 2x y + 1).Thusf(x) + 2f(y) = 2{l1(√y2 + 2x y + 1)+ l2(√y2 + 2x y + 1)}+ f(x y2 + 2 y + xy2 + 2x y + 1)(x, y ∈ U) .Since the functions l1 and l2 are logarithmic, this yields thatf(x) + 2f(y) = l(y2 + 2xy + 1)+ f(x y2 + 2 y + xy2 + 2x y + 1)is fulfilled for any x, y ∈ U , where l = l1 + l2. If we replace here x and y by−x and −y, respectively and subtract the equations side by side,g(x2y + 2x+ y2xy + x2 + 1)= 2g(x) + g(y)can be derived.Thus, if A ∈P2(F) andA =(a bb d),thenA =(√a 00√d)·(1 b√adb√ad1)·(√a 00√d),therefore,φ(A) = 2φ(√a 00√d)+ φ(1 b√adb√ad1)= l1(a) + l2(d) +12l(1− b2ad).Equivalently,φ(A) = l(det(A)) + g(b√ad)+ k(da)Jordan triple mappings 7is satisfied for any A =(a bb d)∈P2(F). Obviously, the mappingP2(F) 3 A 7−→ l(det(A))is a Jordan triple mapping. This means that the functionχ(A) = χ(a bb d)= g(b√ad)+ k(da) (A =(a bb d)∈P2(F))also has to be a Jordan triple mapping. Due to the properties of the functionsg and k,k(da)=χ(a bb d)+ χ(a −b−b d)2,yielding that k is identically zero, in other wordsl1(x) = l2(x) (x ∈]0,+∞[) .All in all, we known that the mapping χ : Pn(F) → R is a Jordan triplemapping andχ(a bb d)= g(b√ad)is valid. From this χ ≡ 0 can de concluded. Definition 6. A function f : P2(F)→ R is called regular if at least one of thefollowing statements are valid.(i) The function f is continuous at a point X0 ∈P2(F).(ii) There exists a set A ⊂ P2(F) of positive Lebesgue measure such thatthe function f is bounded above or below on A;(iii) There exists a set A ⊂ P2(F) of positive Lebesgue measure such thatthe restriction of f to the set A is measurable (in the sense of Lebesgue).The following lemma shows that every logarithmic function ` : ]0,+∞[→R can be expressed with the aid of additive functions, see [8, Theorem 13.1.2.].Lemma 2. Let ` : ]0,+∞[→ R be a logarithmic function. Then there exists anadditive function a : R→ R such that`(x) = a(ln(x)) (x ∈ R) .With the aid of the previous lemma the following statement can bederived for regular logarithmic functions.Theorem 7. Let ` : ]0,+∞[→ R be a regular logarithmic function. Then thereexists κ ∈ R such that`(x) = κ ln(x) (x ∈]0,+∞[) .Theorem 8. Let φ : P2(F)→ R be a mapping for whichφ(Y XY ) = φ(X) + 2φ(Y )8 Eszter Gselmannholds for all X,Y ∈ P2(F). If the function φ is regular, then there existsκ ∈ R such thatφ(X) = κ ln (det(X))holds for all X ∈P2(F).Proof. From the previous theorem we immediately get that there exists alogarithmic function l : ]0,+∞[→ R such thatφ(X) = l(det(X)) (X ∈P2(F)) .Additionally, we know that the function φ is regular. Thus the mappingP2(F) 3 X 7−→ l(det(X))is regular. From this however the regularity of l follows. Thus, there exists aconstant κ ∈ R so thatφ(X) = κ ln(det(X))is satisfied for all X ∈P2(F). 3. The general caseIn view of the results of the previous section, we are able to prove our maintheorem.Theorem 9. Let n ∈ N and assume that for the mapping φ : Pn(F)→ Rφ(Y XY ) = φ(X) + 2φ(Y )holds for all X,Y ∈ Pn(F). Then there is a logarithmic function l : ]0,+∞[→ R so thatφ(X) = l(det(X))is fulfilled for any X ∈Pn(F).Proof. In view of the results of the previous section, the statement of thetheorem is valid for n = 2.The remaining part of the proof will be divided into two parts.Part I. We will show that assuming that the statement is true for acertain n ∈ N, we will show that the statement is also valid for 2n.Let us assume that the statement of the theorem is fulfilled for a certainn ∈ N.Clearly, if the matrices X,Y ∈P2n(F) commute, thenφ(Y XY ) = φ(XY 2),so, the equation for φ simply yields thatφ(XY ) = φ(X) + Φ(Y ),since Φ(Y 2) = 2φ(Y ).Let nowX =(A 00 I)and Y =(I 00 D),Jordan triple mappings 9where A,D ∈Pn(F) and I ∈Pn(F) denotes the n×n identity matrix. ThenX · Y = Y ·Xand since the statement is assumed to be true for n,φ(X) = φ(A 00 I)= l1 (det(A)) ,where l1 : ]0,+∞[→ R is a logarithmic function.Thus, if X ∈P2n(R) has the formX =(A 00 D),where A,D ∈Pn(R), thenφ(X) = φ(A 00 D)= l1(det(A)) + l2(det(D)),where l1, l2 : ]0,+∞[→ R are logarithmic functions.Let now X ∈ Pn(F) be arbitrary, further let A and D stand for theupper and lower n× n minors of X, that is,X =(A BB∗ D)In this case A,D ∈Pn(F).Furthermore,(A BB∗ D)=( √A−1 00 I)·(I 00√D−1)··(I√D−1√A−1√B√A−1√D−1√D−1√A−1√B√A−1√D−1∗I)··(I 00√D−1)·( √A−1 00 I)Therefore, if we determine the function φ on elements having the form(I BB∗ I)(B ∈Mn(F)) ,then φ is completely determined. This can be done in a similar way as in the2× 2 case.Part II. We will show that is the statement is true for n ∈ N, then it isalso true for n− 1.Assume that the statement is valid for a certain n ∈ N, then Pn−1(R)can be embedded to Pn(R) throughPn−1(R) 3 A 7−→(A 00 I)∈Pn(R).10 Eszter GselmannAdditionally, this embedding preserves the Jordan triple product. Define themapping φ˜ : Sn ⊂Pn(R)→ R byφ˜(A 00 1)= φ(A) (A ∈Pn−1(R)) ,whereSn ={(A 00 1)|A ∈Pn−1(R)}.Therefore, if for a mapping φ : Pn−1(R)→ Rφ(Y XY ) = φ(X) + 2φ(Y ) (X,Y ∈Pn−1(R)) ,thenφ(A) = φ˜(A 00 1)= l(det((A 00 1)))= l (1 · det(A)) = l(det(A)),yielding that the statement is valid for n− 1, as well. References[1] Ja´nos Acze´l, Lectures on functional equations and their applications., New Yorkand London: Academic Press. XIX, 510 p. , 1966.[2] Rajendra Bhatia, Positive definite matrices, Princeton Series in Applied Math-ematics, Princeton University Press, Princeton, NJ, 2007. MR 2284176(2007k:15005)[3] Konstancja Bobecka and Jacek Weso lowski, Multiplicative Cauchy functionalequation in the cone of positive-definite symmetric matrices, Ann. Polon. Math.82 (2003), no. 1, 1–7. MR 2041392 (2005a:39049)[4] Mirko Doboviˇsek, Maps from Mn(F) to F that are multiplicative with respectto the Jordan triple product, Publ. Math. Debrecen 73 (2008), no. 1-2, 89–100.MR 2429028 (2009c:15021)[5] Desmond Fearnley-Sander, A characterization of the determinant, Amer. Math.Monthly 82 (1975), no. 8, 838–840. MR 0399126 (53 #2977)[6] Gyula Ga´spa´r, Die Charakterisierung der Determinanten u¨ber einem un-endlichen Integrita¨tsbereich mittels Funktionalgleichungen, Publ. Math. Debre-cen 10 (1963), 244–253. MR 0168584 (29 #5844)[7] Sin Hitotumatu, A characterization of the determinant, Comment. Math. Univ.St. Paul. 13 (1964/1965), 45–50. MR 0178002 (31 #2260)[8] Marek Kuczma, An introduction to the theory of functional equations and in-equalities. Cauchy’s equation and Jensen’s inequality. Edited by Attila Gila´nyi.2nd ed., Basel: Birkha¨user. xiv, 595 p. , 2009.[9] Svetozar Kurepa, On the characterization of the determinant, Glasnik Mat.-Fiz. Astronom. Ser. II Drusˇtvo Mat. Fiz. Hrvatske 19 (1964), 189–198. MR0179187 (31 #3436)[10] B. R. McDonald, A characterization of the determinant, Linear and MultilinearAlgebra 12 (1982/83), no. 1, 31–36. MR 672914 (83j:15009)Jordan triple mappings 11[11] Lajos Molna´r, A remark on the Kochen-Specker theorem and some characteri-zations of the determinant on sets of Hermitian matrices, Proc. Amer. Math.Soc. 134 (2006), no. 10, 2839–2848 (electronic). MR 2231606 (2007b:81018)[12] Kiyosi Yamaguti, Jordan and Jordan triple isomorphisms of rings, J. Sci. Hi-roshima Univ. Ser. A 20 (1956/1957), 107–110. MR 0096710 (20 #3193)Eszter GselmannMTA–DE ’Lendu¨let’Functional Analysis Research GroupInstitute of MathematicsUniversity of DebrecenP. O. Box: 12.DebrecenH–4010Hungarye-mail: gselmann@science.unideb.hu

Jordan triple mappings on positive definite matrices

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Jordan triple mappings on positive definite matrices

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