Maps stemming from the functional calculus that transform a Kubo–Ando mean into another

In this paper, we investigate maps on sets of positive operators which are induced by the continuous functional calculus and transform a Kubo–Ando mean σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} into another τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}. We establish that under quite mild conditions, a mapping ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} can have this property only in the trivial case, i.e. when σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} and τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} are nontrivial weighted harmonic means and ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} stems from a function which is a constant multiple of the generating function of such a mean. In the setting where exactly one of σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} and τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is a weighted arithmetic mean, we show that under fairly weak assumptions, the mentioned transformer property never holds. Finally, when both of σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} and τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} are such a mean, it turns out that the latter property is only satisfied in the trivial case, i.e. for maps induced by affine functions.


Introduction and statement of the results
Homomorphisms of algebraic structures form a basic object of study mainly in algebra, but they are highly relevant also in other areas of mathematics. In general, such transformations on structures of numbers can be described with certain functional equations. The most fundamental ones among them are the Cauchy equations, namely the additive, the multiplicative, the exponential and the logarithmic, which are related to the two most basic operations on real numbers, addition and multiplication. There is a vast literature devoted to them, we mention, e.g. the book of Aczél [1]. Their solutions were determined under very weak regularity assumptions, for example measurability. 762 G. Nagy AEM Homomorphisms are also considered between much more abstract structures than R, e.g. between groups or algebras. For certain algebras, namely C * -algebras, we can define continuous scalar functions also on some of their elements, namely the normal ones. Therefore, from the point of view of homomorphisms, one can consider morphism properties of transformations of the form A → f (A), for example on the set A s of self-adjoint elements of a C *algebra A, where f : R → R is an unknown continuous function. This gives rise to a functional equation for f on an abstract structure, e.g. f (A + B) = f (A) + f (B) (A, B ∈ A s ). It is of the same form as its counterpart on R, the additive Cauchy equation, however, here the value of a function at an element means the value of the continuous functional calculus for that element at the given function. Since the unknown functions in the C * -algebra counterparts of the basic functional equations describing homomorphisms on R are necessarily continuous and these versions imply their validity also on R, one can observe that their solutions can be obtained immediately, applying results on the form of continuous solutions of their scalar counterparts. Therefore their investigation is of no interest.
However, in some other cases, it can give rise to challenging functional equations. For the purposes of the paper, the most important among them is the case where the homomorphisms in question respect certain means, or in other words, transform a mean into another. These quantities form a fundamental concept in mathematics, originally they are introduced for the averaging of real numbers. A mean M : D 2 → D on an interval D ⊂ R is defined as a binary operation satisfying the inequalities min{x, y} ≤ M (x, y) ≤ max{x, y} (x, y ∈ D). Such objects have been intensively studied for a long time by many researchers, their investigation forms a broad field of mathematics. For details on means, the interested reader can consult, e.g. the short monograph [5] and the references therein. As for means of other objects, in [7] Kubo and Ando established the theory of operator means which are certain operations on the cone of positive operators on a Hilbert space. In the finite dimensional case, that notion reduces to means of positive semidefinite matrices which are widely used and investigated in several areas of mathematics.
To define those operator means, we introduce some necessary notation and definitions used throughout the paper. In what follows, H denotes a complex Hilbert space with dim H ≥ 2. Let B(H) stand for the C * -algebra of bounded linear operators on H. An element A ∈ B(H) is termed positive if Ax, x ≥ 0 is satisfied by each vector x ∈ H (which is the same as saying that it is a positive element of the C * -algebra B(H)). Let B(H) + , and B(H) ++ denote the cone of positive operators, and invertible positive operators, resp. in B(H). For a pair A, B of self-adjoint elements in B(H), we write A ≤ B whenever B − A ∈ B(H) + . Finally, we denote by I, the identity operator on H. Vol. 94 (2020) Maps that transform a Kubo-Ando mean into another 763 According to [7], we say that a binary operation σ : B(H) + × B(H) + → B(H) + is a Kubo-Ando mean if it possesses the properties below. For any elements A, B, C, D ∈ B(H) + and sequences (A n ), (B n ) in B(H) + : Here, the symbol ↓ denotes monotone decreasing convergence in the strong operator topology. If σ is a Kubo-Ando mean on B(H) + , then its transposẽ σ : . Kubo-Ando means can be represented by functions of a certain type which are defined as follows. A continuous map f from a nontrivial interval D ⊂ R into R is called n-monotone (or matrix monotone of order n ∈ N) if for each pair A, B of self-adjoint operators on an n-dimensional complex Hilbert space whose spectra are in D, we have the implication We remark that such a function is obviously increasing. If f is n-monotone for any integer n ∈ N, then it is called operator monotone. Moreover, the continuous function f : D → R is called n-concave if for any pair A, B of operators with the above properties the inequality is fulfilled.  Evidently, the property (i) yields F σ (1) = 1. The mentioned proof also shows that if d = dim H < ∞, then F σ is d-monotone, otherwise it is operator monotone. Furthermore, σ is a perspective mean, that is, it admits the explicit form (1) and this, together with the property (iv), yields that F σ uniquely determines σ. DefineF σ := Fσ. By [7, Corollary 4.2], we havẽ In other words, these are the mappings which transform σ to τ . The latter problem seems really challenging. As for related results, first we give the definitions of the three most important classes of Kubo-Ando means, the weighted arithmetic, geometric and harmonic means, whose generating functions on ]0, ∞[ are moreover they are denoted by ∇ α , α and ! α , respectively. They have the explicit forms The structure of automorphisms of B(H) + with respect to them was described in [15,16] for α = 1/2. A result on the automorphisms of B(H) + for any element of a large class of Kubo-Ando means can be found in [14], under some mild regularity assumption (e.g. a sort of continuity) on the transformations in question. As for theorems on maps transforming a Kubo-Ando mean into another, we refer to [13], where the previous three means are considered on the cone of invertible positive elements in a C * -algebra (note that perspective means can be defined on such a cone). Instead of determining the structure of all maps of B(H) + transforming a Kubo-Ando mean into another, we can seek for only those which have a special form. Here, we consider maps of the form Our aim is to establish equivalent conditions under which such a map transforms a Kubo-Ando mean into another. Mappings transforming a mean M of real numbers into another one N are usually termed (M, N )-affine, affine with respect to M and N , or mean affine; they are investigated in, e.g. [9,19]. It is worth noting that (∇ 1/2 , ∇ 1/2 )-affine functions are exactly the solutions of the well-known Jensen equation f ((x + y)/2) = (f (x) + f (y))/2, where f is a real function. Adopting this terminology, for a pair of Kubo-Ando means (σ, τ ) on B(H) + , we say that a continuous function f : Observe that this property says that the map The aim of the paper is to investigate the last displayed functional equation on B(H) + and B(H) ++ . We remark that operator means on the cone of positive definite matrices of a given size are quite important in some areas of mathematics, e.g. they are intimately connected to certain differential geometric structures on those cones. Now our main result follows in which we consider the previous functional equation under some quite mild conditions.
Then the following statements are equivalent: ( As for the conditions in this result, observe that a constant function f satisfies (ii), and f =id fulfills (i) or (ii) if and only if σ = τ . Related to the concavity assumption, observe that the generating function of a Kubo-Ando mean on B(H) + is d-monotone with an appropriate natural number We remark that the condition F σ (0) = 0 means that Aσ0 = 0, whileF σ (0) = 0 means 0σA = 0 (A ∈ B(H) + ). The result above shows the quite surprising fact that, under its conditions, a function is affine with respect to two Kubo-Ando means only in the trivial case.
In certain cases, the property that there exists a function which is affine with respect to two such means can be reformulated in another way. Concerning this, we recall the notion of a conjugated mean. For two means M, N on an interval D ⊂ R, we say that N is a conjugated mean of M if there exists a continuous injective function f : D → D with the property We remark that the terminology refers to group theory. (σ, τ )-affine function. In our next result, we investigate the conjugated means of a given one belonging to any of the sets Corollary. Let i = 1, 2 be a number and σ ∈ M i be a Kubo-Ando mean. Then the only conjugated mean of σ among the elements of M i is σ.
As for the proof, observe that the existence of such a mean M would imply the validity of (ii) in Theorem 1 for M, σ and for the function f appearing in the definition of conjugated means. It is also clear that f, M, σ satisfy the conditions in that result (continuous injective functions are monotone), which would yield M = σ, a contradiction.
Observe that Theorem 1 covers two of the most fundamental classes of Kubo-Ando means, namely the sets of nontrivial weighted geometric and harmonic means, resp., but leave weighted arithmetic means untouched. To fill in this gap, in the next three results, we investigate the conditions under which (i) or (ii) in Theorem 1 holds for two Kubo-Ando means, at least one of them being a weighted arithmetic mean. The latter two theorems assert that for the members of a large class of Kubo-Ando means, there does not exist nonconstant functions which are affine with respect to one of them and to a nontrivial weighted arithmetic mean. Now our last result follows in which we establish that for means of the latter kind, the only mean affine functions are the trivial, affine ones.
exactly when we have scalars a, b ∈ R such that f (x) = ax + b (x > 0) and in the case a = 0, the equality α = β holds.
It is worth mentioning that Theorem 3 says that there are no nontrivial weighted arithmetic means which are conjugated means of some element of M 1 ∪ M 2 . As for the conjugation of other Kubo-Ando means, it follows from [17,Theorem 1.4.] that such an operation on B(H) + is a weighted quasiarithmetic mean (i.e., a conjugated mean of a weighted arithmetic mean) exactly when it is a weighted arithmetic mean. In other words, the only Kubo-Ando means that are conjugated means of ∇ α (0 ≤ α ≤ 1) are the ones of this form, so only itself, by Theorem 4.

Proofs
Rank-one (orthogonal) projections on H will show up several times in this section, P 1 (H) stands for their class. The members of P 1 (H) are exactly the operators of the form u⊗u (u ∈ H, u = 1), where the operation ⊗ is defined by In this section, we will use the well-defined quantity where A ∈ B(H) ++ , P ∈ P 1 (H) are elements and u ∈ rng P is a unit vector, moreover rng denotes the range of operators. Now we are in a position to verify our main result.
Proof of Theorem 1. The implications (ii) =⇒ (i), (iii) =⇒ (ii) are trivial. Assume that (i) holds. In what follows, we show that, without loss of generality, F σ (0) = F τ (0) = 0 may be supposed. By [18, Lemma 1.3.2], a function g from an interval D ⊂ R to R is strictly concave if and only if for any numbers

G. Nagy AEM
It can be checked easily that for any Kubo-Ando mean ρ and each scalars 0 < x 1 < x 2 < x 3 , the equality 1 holds. This gives us the strict concavity ofF σ ,F τ and then, sinceF σ = Fσ,F τ = Fτ , it follows that the relation F σ (0) = F τ (0) = 0 can and will be assumed. Next, observe that by (1) Furthermore if (ii) holds, then by the discussion in the previous paragraph, we may and do suppose that F σ (0) = F τ (0) = 0.
To sum up, we can conclude that both (i) and (ii) imply (5) and that F σ (0) = F τ (0) = 0 can and will be assumed. Hence it remains to show that (5) and the relations F σ (0) = F τ (0) = 0 yield (iii). To prove it, observe that from applying the property (iii) of Kubo-Ando means (see Sect. 1) their homogeneity follows, thus after multiplying (5) by f (1) −1 , wlog f (1) = 1 may and will be supposed. Now using a proof by contradiction, we verify that 1}, furthermore f (0)/f (x 0 ) = 0, 1; the second last equation would yield the equality of the strictly concave function F τ and of an affine one at three distinct points, a contradiction. We proceed by inserting an arbitrary element B = P = u⊗u (u ∈ H, u = 1) in (5). To get the desired conclusion from this substitution, we need a formula for AρP , where A ∈ B(H) ++ is an operator and ρ is a Kubo-Ando mean with F ρ (0) = 0. This can be derived by the computation Vol. 94 (2020) Maps that transform a Kubo-Ando mean into another 769 from which we conclude that AρP =F ρ (λ(A, P ))P. Now applying (5) for B = P and this formula (observe that A ∈ B(H) ++ and f (x) > 0 (x > 0), so f (A) ∈ B(H) ++ ), because of the equalities f (0) = 0, f(1) = 1, it follows that f (F σ (λ(A, P )))P =F τ (λ(f (A), P ))P , i.e.
Next put A = xI, B = yI (x, y > 0) in (5) to get the relation which, for y = 1, gives us that f (F σ (x)) =F τ (f (x)). This together with (6) impliesF Observe that sinceF τ is strictly concave and monotone, it is injective. This fact together with the last displayed equality gives us that f (λ(A, P )) = λ(f (A), P ). It implies Now let e 1 , e 2 ∈ H; x, y > 0; 0 < μ < 1 be arbitrary elements such that e 1 , e 2 are mutually orthogonal unit vectors, and plug A = x · e 1 ⊗ e 1 + y · e 2 ⊗ e 2 + I − (e 1 ⊗ e 1 + e 2 ⊗ e 2 ), in the previous equality to obtain, using the basic properties of functional calculus, that In the rest of the proof, for any function g : To complete the proof, we have to show that F σ , F τ are of the form appearing in (iii). To do this, let x, y > 0 be arbitrary numbers and substitute the latter form of f in (7) in order to get that where the left-hand side equalŝ therefore, with the notation t = 1/y, the latter equality yields that Next, let 0 < ν < 1 be an arbitrary number. It is clear that we can find a scalar x ν > 0 for which ν = (1 − β)/((1 − β) + βx ν ). Then it follows that for all t > 0, the equationF τ (ν(  (8) with respect to ν, we arrive at the relation (s − 1)F τ (νs + 1 − ν) =F σ (s) − 1 for any given s > 0. By the latter facts, we can take the limit ν → 0 in this equality and then obtain that (s − 1)F τ (1) =F σ (s) − 1, from which we deduce the relationF σ (s) = γs + 1 − γ with γ =F τ (1). Now we conclude that F σ (x) = (γx −1 + 1 − γ) −1 (x > 0) and, because of the conditions on F σ , the inequalities 0 < γ < 1 follow. Finally, substituting the previous form ofF σ in (8) and passing to the limit ν → 1 in the relation obtained so, we arrive at the formulâ F τ (s) = γs + 1 − γ (s > 0) entailing the desired equality F τ = F σ . By what we have proved so far and having in mind how we have reached the assumptions F σ (0) = F τ (0) = 0, f(1) = 1; the condition (iii) follows and this completes the proof of Theorem 1. This means that f, g, h satisfy the Pexiderized Cauchy additive functional equation on the domain ∅ = ]0, ∞[ × ]0, ∞[ ⊂ R 2 , therefore Theorem 4 in [1, p. 80] applies yielding the existence of a function ϕ : R → R and constants b 1 , b 2 ∈ R with the properties that ϕ fulfills the Cauchy additive equation and f = ϕ + b 1 + b 2 , g = ϕ + b 1 , h = ϕ + b 2 . However, the continuity of f implies that of ϕ and thus we immediately obtain that one can find a number a ∈ R for which f (x) = ax + b 1 + b 2 = ((aα)/β)x + b 1 /β, hence a = a(α/β), and if a = 0, then α = β. The statement of Theorem 4 follows.

Concluding remarks
As far as we know, the structure of homomorphisms of B(H) + or B(H) ++ with respect to a Kubo-Ando mean is still unknown in full generality even in the bijective case. Here we have obtained partial results about this problem