A Survey on Operator Monotonicity , Operator Convexity , and Operator Means

This paper is an expository devoted to an important class of real-valued functions introduced by Löwner, namely, operator monotone functions. This concept is closely related to operator convex/concave functions. Various characterizations for such functions are given from the viewpoint of differential analysis in terms of matrix of divided differences. From the viewpoint of operator inequalities, various characterizations and the relationship between operator monotonicity and operator convexity are given by Hansen and Pedersen. In the viewpoint of measure theory, operator monotone functions on the nonnegative reals admit meaningful integral representations with respect to Borel measures on the unit interval. Furthermore, Kubo-Ando theory asserts the correspondence between operator monotone functions and operator means.


Introduction
A useful and important class of real-valued functions is the class of operator monotone functions.Such functions were introduced by Löwner in a seminal paper [1].These functions are functions of Hermitian matrices/operators preserving order.In that paper, he established a relationship between operator monotonicity, the positivity of matrix of divided differences, and an important class of analytic functions, namely, Pick functions.This concept is closely related to operator convex/concave functions which was studied afterwards by Kraus in [2].Operator monotone functions and operator convex/concave functions arise naturally in matrix and operator inequalities (e.g., [3][4][5][6][7]).This is because the theory of inequalities depends heavily on the concepts of monotonicity, convexity, and concavity.One of the most beautiful and important results in operator theory is the so-called Löwner-Heinz inequality (see [1,8]) which is equivalent to the operator monotonicity of the function   →   for  ⩾ 0 when  ∈ [0, 1].See more information about operator monotonicity/convexity in [5, Chapter V], [9, Section 2], [10,Chapter 4], and [11].
Operator monotone functions have applications in many areas, including functional analysis, mathematical physics, information theory, and electrical engineering; see, for example, [12][13][14][15].This concept plays major roles in the so-called Kubo-Ando theory of operator connections and operator means.This axiomatic theory was introduced in [16] and plays central role in operator inequalities, operator equations, network theory, and quantum information theory.Indeed, there is a one-to-one correspondence between operator monotone functions on the nonnegative reals R + and operator connections.See more information about applications of operator monotone functions to theory of operator means in [9,10,[17][18][19].
In this paper, we survey significant results of operator monotone functions related to operator convexity and operator means.We give various characterizations in several viewpoints.The first viewpoint is differential analysis in terms of matrix of divided differences.In viewpoint of operator inequalities, Hansen and Pedersen provide characterizations and relationship of operator monotonicity and operator convexity/concavity.From measure theory viewpoint, every operator monotone function on the nonnegative reals always occurs as an integral of suitable operator monotone functions with respect to a Borel measure.Such functions form building blocks for arbitrary operator monotone functions on R + .A deep theory of Kubo and Ando states that each operator 2 International Journal of Analysis monotone function  on R + corresponds to a unique operator connection.Moreover, if (1) = 1, then  is associated with an operator mean.
Here is the outline of the paper.In Section 2, after setting basic notations, we give the definitions and examples of operator monotone/convex functions and provide their characterizations with respect to matrix of divided differences.Section 3 deals with Hansen-Pedersen characterizations of operator monotone/convex functions.The aims of Section 4 are to characterize operator monotone functions on the nonnegative reals in terms of Borel measures and to give some concrete examples.Finally, we establish the correspondence between operator monotone functions and operator means in Section 5.

Operator Monotonicity and Convexity
Throughout this paper, let H be a complex Hilbert space.Denote by (H) the algebra of bounded linear operators on H.The spectrum of an operator  is denoted by ().Consider the real vector space (H) sa of self-adjoint operators on H and its positive cone (H) + of positive operators on H.A partial order is naturally equipped on (H) sa by defining  ⩽  if and only if  −  ∈ (H) + .We write  > 0 to mean that  is a strictly positive operator, or equivalently,  ⩾ 0 and  is invertible.When H = C  , we identify (H) with the algebra M  of -by- complex matrices.Then M +  is just the cone of -by- positive semidefinite matrices.
Let  ∈ M  be normal.By the spectral resolution of , there exist distinct scalars  1 , . . .,   ∈ C and projections  1 , . . .,   on C  such that Moreover, these scalars and projections are uniquely determined.In fact, () is the set { 1 , . . .,   } (not counting multiplicities) and each   is the projection onto the eigenspace ker( −   ).For each function  : () → C, we can define the functional calculus for the function  by Every -monotone function is ( − 1)-monotone but the converse is false in general.The condition of being 1monotone is the monotone increasing in usual sense.The set of operator monotone functions on the interval  is closed under taking nonnegative linear combinations, pointwise limits, and compositions.The straight line   →  +  is operator concave and operator convex on the real line for any ,  ∈ R.This function is operator monotone if and only if the slope  is nonnegative.Proposition 2. On (0, ∞), the function   → 1/ is operator convex and   → −1/ is operator monotone.On (−∞, 0), the function   → 1/ is operator concave and   → −1/ is operator monotone.
It follows from this proposition that the function   → (− ) −1 is operator monotone on (, ) for any  ∉ (, ).The next result is called the Löwner-Heinz inequality.It was first proved by Löwner [1] and also by Heinz [8].There are many proofs of this fact.The following is due to Pedersen [20].
Operator monotone functions can be defined in the context of operators acting on a Hilbert space as illustrated in the next theorem.This is why we also call a matrix monotone function an operator monotone function.Note that in this theorem we assume the continuity of  since we need to define the continuous functional calculus of an operator.Theorem 5.The following statements are equivalent for a continuous function  : (, ) → R: where (    ) is the functional calculus of     in ().
Since () is identified with M  with  = dim and since     ⩽     as elements of (), the assumption (i) implies that (    ) ⩽ (    ) and hence (  ) ⩽ (  ).By taking the limit in the strong-operator topology, we have () ⩽ ().

Differential Analysis of Operator Monotonicity and Convexity
In this section, we consider topological properties of operator monotone/convex functions.Note that we do not impose a topological assumption on any -monotone/concave function.The following three theorems show that any -monotone/concave function on an open interval is at least continuously differentiable (i.e.,  1 ) function when  ⩾ 2.
Theorem 6.If  is a 2-monotone function on (, ), then  is  1 on (, ) and   > 0 unless  is a constant.In particular, every operator monotone function on (, ) is  1 .
Proof.The proof is very long and it consists of many details.The original proof is contained in [1]; see also [9,Section 2].

Hansen-Pedersen Characterizations
In this section, we characterize operator monotone functions in the sense of Hansen-Pedersen [21].
(iii) ⇔ (iv).Write () = 1/().Let ,  > 0 in M  .By (iii), Then Proposition 2 implies Hence  is operator convex.Example 14.In information theory, the function () = − log  is known as the entropy function.In [22], it was shown that this function is operator concave.An analogue notion of the entropy function in quantum mechanics is the entropy of a density matrix (a positive semidefinite matrix with trace 1) or a positive contraction on a Hilbert space.More precisely, for each positive operator  with ‖‖ ⩽ 1, we define the entropy of  by More concrete examples of operator monotone functions are provided in [23].

Integral Representations of Operator Monotone Functions on the Nonnegative Reals
In this section, we focus on the class of operator monotone functions from R + to R + , denoted by OM(R + ).A reformulation of Löwner's theorem (see [1] or [5, Chapter V]) states that every  ∈ OM(R + ) admits an integral representation with respect to Borel measure on the unit interval as follows.
Theorem 15 (see [24]).Given a finite Borel measure  on [0, 1], the function is an operator monotone function from R + to R + .In fact, the map   →  is bijective, affine, and order-preserving.
Thus the functions   → 1!   for  ∈ [0, 1] form building blocks for arbitrary operator monotone functions on R + .The measure  in the previous theorem is called the associated measure of the operator monotone function .Moreover, a function  ∈ OM(R + ) is normalized (in the sense that (1) = 1) if and only if  is a probability measure [24].This means that every normalized operator monotone function on R + can be viewed as an average of the special operator monotone functions   → 1!   for  ∈ [0, 1].The functions   → 1!   for  ∈ [0, 1] are extreme points of the convex set of normalized operator monotone functions from R + to R + .
The following examples illustrate the associated measures of "absolutely continuous" operator monotone functions; see [24] for details of proofs.
(1) For each 0 <  < 1, the associated measure of the operator monotone function   is given by (2) The associated measure of () = ( − 1)/ log  is () = () where density function  given by (3) Consider the operator monotone function This function has Lebesgue measure as the associated measure, equivalently; we have the integral representation If  ∈ OM(R + ) has  as the associated measure, then the transpose of , defined by   → (1/), also belongs to OM(R + ) and has Θ as its associated measure.Here, Θ : [0, 1] → [0, 1] is a homeomorphism defined by   → 1 − .
We say that  is symmetric if it coincides with its transpose.A Borel measure  on [0, 1] is said to be symmetric if  is invariant under Θ; that is, Θ = .

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Corollary 18 (see [24]).There is a one-to-one correspondence between symmetric operator monotone function from R + to R + and finite symmetric Borel measures via the integral representation: In particular, an operator monotone function on R + is symmetric if and only if its associated measure is symmetric.
It follows that there is a one-to-one correspondence between normalized symmetric operator monotone functions on R + and probability symmetric Borel measures on the unit interval via the integral representation (29).
The integral representation (24) also has advantages in treating decompositions of operator monotone functions (see [24]).It turns out that every function  ∈ OM(R + ) can be expressed as where  ac ,  sd , and  sc also belong to the class OM(R + ).The "singularly discrete part"  sd is a countable sum of   → 1!   for  ∈ [0, 1] with nonnegative coefficients.The "absolutely continuous part"  ac has an integral representation with respect to Lebesgue measure  on [0, 1].The "singularly continuous part"  sc has an integral representation with respect to a continuous measure mutually singular to .

Operator Monotone Functions and Operator Means
This section explains the one-to-one correspondence between operator monotone functions on R + and operator means.
An axiomatic of operator means was investigated by Kubo and Ando [16].Recall that an operator connection is a binary operation  on (H) + such that for all positive operators , , , : An operator mean is an operator connection  with property that  =  for all  ⩾ 0, or, equivalently,  = 0.
From the monotonicity (M1) and the continuity (M3), every operator connection is uniquely determined on the set of strictly positive operators.Indeed, once an operator connection  is defined for any ,  > 0, we have  = lim ↓0 ( + )  ( + ) ; (31) here the limit is taken in the strong-operator topology.
A major core of Kubo-Ando theory is the one-to-one correspondence between operator connections and operator monotone functions.
Theorem 19 (see [16,Theorem 3.4]).Given an operator connection , there is a unique operator monotone function  : R + → R + such that  () = ,  ⩾ 0. (32) In fact, the map   →  is a bijection.Moreover,  is a mean if and only if  is a probability measure.
The function  in this theorem is called the representing function of the operator connection .It turns out that, for any ,  > 0, From the integral representation of operator monotone functions (24), every operator connection admits the integral representation Since the weighted harmonic means are jointly concave, it follows that every operator connection is jointly concave.
Example 20.The function () = 2/(1 + ) is operator monotone on R + according to Corollary 12.This function gives rise to the harmonic mean: Moreover, ! is the largest positive operator  such that (see [3,Theorem I.3]).
Example 21.For each  ∈ (0, 1), the function   () =   is operator monotone on R + by the Löwner-Heinz inequality (Theorem 3).Each   corresponds to the -weighted geometric mean: Let us have a close look for the geometric mean # 1/2 , denoted briefly by #.In the literature, there are various characterizations of the geometric mean.This mean was firstly defined by Pusz and Woronowicz [25]: This definition coincides with the following formula given by Ando [3]: Alternatively, the geometric mean of ,  > 0 is the common limit of the following iterative process (see [26]): It was also pointed out in [14] that the geometric mean of ,  > 0 is the unique positive solution to the Riccati equation: When  = 0, it is understood that we take limit as  tends to 0 and, by L'Hôspital's rule,  0, () =   .
Each function  , gives rise to a unique operator mean, namely, the quasiarithmetic power mean # , with exponent  and weight  as follows: The family of quasiarithmetic power means includes the weighted arithmetic means (the case  = 1), the weighted harmonic means (the case  = −1), and the weighted geometric means (the case  = 0) as special cases.
Example 23.Let  ∈ [−1, 1] and consider the operator monotone function (see [27]) Since each   is symmetric and normalized, it associates to a unique symmetric operator mean.Note that the family of these means include the arithmetic mean, the geometric mean, and the harmonic mean.The cases  = 1/3 and  = −1/3 are known as the logarithmic mean and its dual.
The case  = 1 gives rise to the operator mean, called the identric mean.The case  = 0 is associated to the logarithmic mean.
The relative operator entropy is closely related to the Karcher mean theory.The Karcher mean or the weighted leastsquares mean of the tuple  = ( 1 ,  2 , . . .,   ) of strictly positive operators with weights  = ( 1 ,  2 , . . .,   ) of positive real numbers such that ∑   = 1 is geometrically defined to be arg min See more information about Karcher mean theory in [31][32][33][34] and references therein.