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

Chansangiam, Pattrawut

doi:https://doi.org/10.1155/2015/649839

International Journal of Analysis

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Research Article | Open Access

Volume 2015 | Article ID 649839 | https://doi.org/10.1155/2015/649839

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

Pattrawut Chansangiam¹

Academic Editor: Julien Salomon

Received31 Jul 2015

Accepted18 Oct 2015

Published11 Nov 2015

Abstract

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.

1. 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., [3–7]). 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 when . 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, [12–15]. 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 and operator connections. See more information about applications of operator monotone functions to theory of operator means in [9, 10, 17–19].

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 . A deep theory of Kubo and Ando states that each operator monotone function on corresponds to a unique operator connection. Moreover, if , 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.

2. Operator Monotonicity and Convexity

Throughout this paper, let be a complex Hilbert space. Denote by the algebra of bounded linear operators on . The spectrum of an operator is denoted by . Consider the real vector space of self-adjoint operators on and its positive cone of positive operators on . A partial order is naturally equipped on by defining if and only if . We write to mean that is a strictly positive operator, or equivalently, and is invertible. When , we identify with the algebra of -by- complex matrices. Then is just the cone of -by- positive semidefinite matrices.

Let be normal. By the spectral resolution of , there exist distinct scalars and projections on such thatMoreover, these scalars and projections are uniquely determined. In fact, is the set (not counting multiplicities) and each is the projection onto the eigenspace . For each function , we can define the functional calculus for the function by

Definition 1. Let be an interval. A function is said to be as follows:(i) Matrix monotone of degree or -monotone if, for every with , it holds that(ii) Operator monotone or matrix monotone if it is -monotone for all .(iii) Matrix convex of degree or -convex if, for every with , it holds that(iv) Operator convex if it is -convex for every .(v)Matrix concave of degree or -concave if is -convex.(vi) Operator concave if it is -concave for every .

Recall that a continuous function is convex (concave) if and only if it is midpoint-convex (midpoint-concave, resp.). It follows that a continuous function is -convex if and only if it is -midpoint convex; that is, the condition (4) holds for . In particular, if is continuous, then is operator convex if and only if it is operator midpoint-convex. Similar results are applied for the case of concavity.

Every -monotone function is -monotone but the converse is false in general. The condition of being -monotone 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 . This function is operator monotone if and only if the slope is nonnegative.

Proposition 2. On , the function is operator convex and is operator monotone. On , the function is operator concave and is operator monotone.

Proof. If , then and hence . This shows the operator monotonicity of on . The scalar inequality implies that for every For in , by setting ; we haveHence is operator convex. For the case , consider and instead of and .

It follows from this proposition that the function 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].

Theorem 3. For in and , one has .

Proof. The continuity argument allows us to consider . Since and are continuous, the set is closed. Clearly, . Recall that the set of dyadic numbers in is dense in . Hence, to prove that , it suffices to show thatSuppose and . Then andHence, and similarly . Thus Here, denotes the spectral radius. Now, or ; that is, .

Proposition 4. For each , the function is not operator monotone on .

Proof. Consider and . Then . Since is a projection, for each , we have andCompute If , we must have ; that is, , which is false when .

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 :(i) for all Hermitian matrices of all orders whose spectra are contained in ;(ii) for all Hermitian operators whose spectra are contained in and for an infinite-dimensional Hilbert space ;(iii) for all Hermitian operators whose spectra are contained in and for all Hilbert spaces .

Proof. It is obvious that (iii) implies (ii). The implication (ii) (i) follows by taking an -dimensional subspace.
(i) (iii). For each finite-dimensional subspace of , let be the orthogonal projection onto . Suppose that in with spectra in . Consider nets and in , where is fixed and a directed set:with respect to the set inclusion. Since and in the strong-operator topology, we have and in the strong-operator topology. Note that where is the functional calculus of in . Since is identified with with and since as elements of , the assumption (i) implies that and hence . By taking the limit in the strong-operator topology, we have .

3. 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., ) function when .

Theorem 6. If is a -monotone function on , then is on and unless is a constant. In particular, every operator monotone function on is .

Proof. The proof is very long and it consists of many details. The original proof is contained in [1]; see also [9, Section 2].

Theorem 7. Let be an integer. The following statements are equivalent for a function :(1) is -monotone on ;(2) is on and for every choice of from .Here, the 1st divided difference is defined as follows:and .

Proof. See [1] or [9, Section 2].

Theorem 8. Let be an integer. The following statements are equivalent for a function :(1) is -convex on ;(2) is on and for every choice of from .Moreover, if is operator convex, then is operator monotone for every . Here, the 2nd divided difference is defined to beand .

Proof. See [2] or [9, Section 2].

4. Hansen-Pedersen Characterizations

In this section, we characterize operator monotone functions in the sense of Hansen-Pedersen [21].

Theorem 9. Let be a function where . Then the following are equivalent:(i) is operator convex on and ;(ii) is operator convex on and , where the existence of and are automatic from the operator convexity of on ;(iii) is operator monotone on and , where the existence of and are automatic from the operator monotonicity of on ;(iv) for every with , for every with and for every ;(v) for every with , for every , with and for every ;(vi) for every with , for every orthogonal projection on and for every .

Proof. See [21] or [9, Theorem 2.5.2].

Theorem 10. If and for all , then the conditions of Theorem 9 are also equivalent to(vii) being operator monotone on .

Proof. See [21] or [9, Theorem 2.5.3].

Corollary 11. A function on is operator monotone if and only if is operator concave.

Proof. This is the equivalence between (i) and (vii) of Theorem 10.

Corollary 12. Consider the following statements for a function :(i) is operator monotone;(ii) is operator monotone;(iii) is operator concave;(iv) is operator convex.We have (i) (ii) (iii) (iv).

Proof. (i) (ii). For any , is operator monotone on . Theorem 10 implies that is operator monotone on . Proposition 2 then implies thatis operator monotone on . Letting yields (ii).
(ii) (i). For any , is operator monotone on . Theorem 10 implies that is operator monotone on . Proposition 2 then implies thatis operator monotone on . Letting yields (i).
(i) (iii). By Corollary 11, we have that(iii) (iv). Write . Let in . By (iii),Then Proposition 2 impliesHence is operator convex.

Example 13. Consider the following:(i)For each , is operator concave on .(ii)The function on where and .(iii)The logarithmic function is operator monotone and operator concave on .(iv)The function is operator convex on .

Proof. (i) It follows from the Löwner-Heinz inequality and Corollary 11. (ii) Use the integral representation for . (iii) By (ii), is operator monotone function on . Corollary 12 then implies that is operator monotone and operator concave on . Now, for each , is operator monotone and operator concave on . Letting yields the result. (iv) Since is continuous on and is operator monotone on , we can conclude that is operator convex on by Theorem 9.

Example 14. In information theory, the function 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 ) or a positive contraction on a Hilbert space. More precisely, for each positive operator with , we define the entropy of by

More concrete examples of operator monotone functions are provided in [23].

5. Integral Representations of Operator Monotone Functions on the Nonnegative Reals

In this section, we focus on the class of operator monotone functions from to , denoted by . A reformulation of Löwner’s theorem (see [1] or [5, Chapter V]) states that every 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 , the functionis an operator monotone function from to . In fact, the map is bijective, affine, and order-preserving.

Thus the functions for form building blocks for arbitrary operator monotone functions on . The measure in the previous theorem is called the associated measure of the operator monotone function . Moreover, a function is normalized (in the sense that ) if and only if is a probability measure [24]. This means that every normalized operator monotone function on can be viewed as an average of the special operator monotone functions for . The functions for are extreme points of the convex set of normalized operator monotone functions from to .

Example 16 (typical examples of “singularly discrete” operator monotone functions). The operator monotone function corresponds to the Dirac measure at . In particular, the operator monotone functions and correspond to the measures and , respectively. By affinity of the map , the measure , where and , is associated to the function .

Example 17. The following examples illustrate the associated measures of “absolutely continuous” operator monotone functions; see [24] for details of proofs.(1)For each , the associated measure of the operator monotone function is given by(2)The associated measure of 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 has as the associated measure, then the transpose of , defined by , also belongs to and has as its associated measure. Here, is a homeomorphism defined by .

We say that is symmetric if it coincides with its transpose. A Borel measure on is said to be symmetric if is invariant under ; that is, .

Corollary 18 (see [24]). There is a one-to-one correspondence between symmetric operator monotone function from to and finite symmetric Borel measures via the integral representation:In particular, an operator monotone function on 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 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 can be expressed as where , , and also belong to the class . The “singularly discrete part” is a countable sum of for with nonnegative coefficients. The “absolutely continuous part” has an integral representation with respect to Lebesgue measure on . The “singularly continuous part” has an integral representation with respect to a continuous measure mutually singular to .

6. Operator Monotone Functions and Operator Means

This section explains the one-to-one correspondence between operator monotone functions on 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 such that for all positive operators :(M1)monotonicity: , ;(M2)transformer inequality: ;(M3)continuity from above: for , if and , then . Here, indicates that is a decreasing sequence converging strongly to .An operator mean is an operator connection with property that for all , or, equivalently, .

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 , we have 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 such thatIn 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 ,From the integral representation of operator monotone functions (24), every operator connection admits the integral representationSince the weighted harmonic means are jointly concave, it follows that every operator connection is jointly concave.

Example 20. The function is operator monotone on 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 , the function is operator monotone on by the Löwner-Heinz inequality (Theorem 3). Each corresponds to the -weighted geometric mean:Let us have a close look for the geometric mean , 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 is the common limit of the following iterative process (see [26]):It was also pointed out in [14] that the geometric mean of is the unique positive solution to the Riccati equation:Moreover, the geometric mean is the largest self-adjoint operator for which(see [3, Theorem I.2]). See more information of the geometric mean in [6, Section 4.1] and [10, Section 5.1].

Example 22. For each and , consider the operator monotone function (see e.g., [5]):When , it is understood that we take limit as tends to and, by L’Hôspital’s rule, 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 ), the weighted harmonic means (the case ), and the weighted geometric means (the case ) as special cases.

Example 23. Let 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 and are known as the logarithmic mean and its dual.

Example 24. For each , the functionbelongs to . Each gives rise to an operator mean, called the Heinz mean. See more information in [28].

Example 25. For each , consider the operator monotone functionwhere the case is understood asThe case gives rise to the operator mean, called the identric mean. The case is associated to the logarithmic mean.

Example 26. The function is operator monotone on ; see, for example, [5]. Fujii and Kamei [29] introduced the relative operator entropy to be the operator connection associated with this function by (33) as follows:Note that for any . The relative operator entropy satisfies attractive entropy-like properties in physical literature (see also [30]).
The relative operator entropy is closely related to the Karcher mean theory. The Karcher mean or the weighted least-squares mean of the tuple of strictly positive operators with weights of positive real numbers such that is geometrically defined to beHere, is a Riemannian metric given byIt turns out that the Karcher mean (51) is the unique positive definite solution to the equationSee more information about Karcher mean theory in [31–34] and references therein.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank referees for valuable suggestions. The author was supported by King Mongkuts Institute of Technology Ladkrabang Research Fund Grant no. KREF045710.

References

C. Löwner, “Über monotone matrix funktionen,” Mathematische Zeitschrift, vol. 38, pp. 177–216, 1934.
View at: Google Scholar
F. Kraus, “Über konvexe Matrixfunktionen,” Mathematische Zeitschrift, vol. 41, no. 1, pp. 18–42, 1936.
View at: Publisher Site | Google Scholar | MathSciNet
T. Ando, Topics on Operator Inequalities, Hokkaido University, Sapporo, Japan, 1978.
View at: MathSciNet
T. Ando, “Concavity of certain maps on positive definite matrices and applications to Hadamard products,” Linear Algebra and its Applications, vol. 26, pp. 203–241, 1979.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Bhatia, Matrix Analysis, Springer, New York, NY, USA, 1996.
R. Bhatia, Positive Definite Matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, USA, 2007.
View at: MathSciNet
X. Zhan, Matrix Inequalities, Springer-Verlag, New York, NY, USA, 2002.
View at: Publisher Site | MathSciNet
E. Heinz, “Beiträge zur Störungstheorie der Spektralzerlegung,” Mathematische Annalen, vol. 123, pp. 415–438, 1951.
View at: Publisher Site | Google Scholar | MathSciNet
F. Hiai, “Matrix analysis: matrix monotone functions, matrix means, and majorization,” Interdisciplinary Information Sciences, vol. 16, no. 2, pp. 139–248, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
F. Hiai and D. Petz, Introduction to Matrix Analysis and Applications, Springer-Verlag, New Delhi, India, 2014.
View at: Publisher Site | MathSciNet
F. Hiai and K. Yanagi, Hilbert Spaces and Linear Operators, Makino Publishing, 1995.
W. N. Anderson Jr. and G. E. Trapp, “A class of monotone operator functions related to electrical network theory,” Linear Algebra and Its Applications, vol. 15, no. 1, pp. 53–67, 1976.
View at: Publisher Site | Google Scholar | MathSciNet
W. N. Anderson Jr., T. D. Morley, and G. E. Trapp, “Positive solutions to $= A - {B X}^{- 1} B$ ,” Linear Algebra and Its Applications, vol. 134, pp. 53–62, 1990.
View at: Publisher Site | Google Scholar
W. N. Anderson Jr. and G. E. Trapp, “Operator means and electrical networks,” in Proceedings of the IEEE International Symposium on Circuits and Systems, pp. 523–527, 1980.
View at: Google Scholar
E. P. Wigner and J. von Neumann, “Significance of Löwner's theorem in the quantum theory of collisions,” Annals of Mathematics, vol. 59, pp. 418–433, 1954.
View at: Publisher Site | Google Scholar | MathSciNet
F. Kubo and T. Ando, “Means of positive linear operators,” Mathematische Annalen, vol. 246, no. 3, pp. 205–224, 1980.
View at: Publisher Site | Google Scholar | MathSciNet
P. Chansangiam and W. Lewkeeratiyutkul, “Characterizations of connections for positive operators,” Southeast Asian Bulletin of Mathematics, vol. 37, no. 5, pp. 645–657, 2013.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. Chansangiam, “Operator means and applications,” in Linear Algebra: Theorems and Applications, H. A. Yasser, Ed., chapter 8, InTech, Rijeka, Croatia, 2012.
View at: Publisher Site | Google Scholar
P. Chansangiam, “Positivity, betweenness, and strictness of operator means,” Abstract and Applied Analysis, vol. 2015, Article ID 851568, 5 pages, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
G. K. Pedersen, “Some operator monotone functions,” Proceedings of the American Mathematical Society, vol. 36, pp. 309–310, 1972.
View at: Google Scholar | MathSciNet
F. Hansen and G. K. Pedersen, “Jensen's inequality for operators and Löwner's theorem,” Mathematische Annalen, vol. 258, no. 3, pp. 229–241, 1982.
View at: Publisher Site | Google Scholar | MathSciNet
M. Nakamura and H. Umegaki, “A note on the entropy for operator algebras,” Proceedings of the Japan Academy, vol. 37, pp. 149–154, 1961.
View at: Publisher Site | Google Scholar | MathSciNet
T. Furuta, “Concrete examples of operator monotone functions obtained by an elementary method without appealing to Löwner integral representation,” Linear Algebra and its Applications, vol. 429, no. 5-6, pp. 972–980, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
P. Chansangiam, “Integral representations and decompositions of operator monotone functions on the nonnegative reals,” http://arxiv.org/abs/1304.7936v1.
View at: Google Scholar
W. Pusz and S. L. Woronowicz, “Functional calculus for sesquilinear forms and the purification map,” Reports on Mathematical Physics, vol. 8, no. 2, pp. 159–170, 1975.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. N. Anderson Jr., T. D. Morley, and G. E. Trapp, “Characterization of parallel subtraction,” Proceedings of the National Academy of Sciences of the United States of America, vol. 76, no. 8, pp. 3599–3601, 1979.
View at: Google Scholar
J. I. Fujii and Y. Seo, “On parametrized operator means dominated by power ones,” Scientiae Mathematicae, vol. 1, no. 3, pp. 301–306, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. Bhatia, “Interpolating the arithmetic-geometric mean inequality and its operator version,” Linear Algebra and its Applications, vol. 413, no. 2-3, pp. 355–363, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
J. Fujii and E. Kamei, “Relative operator entropy in noncommutative information theory,” Mathematica Japonica, vol. 34, no. 3, pp. 341–348, 1989.
View at: Google Scholar | MathSciNet
J. I. Fujii, “Operator means and the relative operator entropy,” in Operator Theory and Complex Analysis, vol. 59 of Operator Theory: Advances and Applications, pp. 161–172, Birkhäuser, Basel, Switzerland, 1992.
View at: Publisher Site | Google Scholar
J. Lawson and Y. Lim, “Monotonic properties of the least squares mean,” Mathematische Annalen, vol. 351, no. 2, pp. 267–279, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
J. Lawson and Y. Lim, “Weighted means and Karcher equations of positive operators,” Proceedings of the National Academy of Sciences of the United States of America, vol. 110, no. 39, pp. 15626–15632, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
J. D. Lawson and Y. Lim, “Karcher means and Karcher equations of positive definite operators,” Transactions of the American Mathematical Society Series B, vol. 1, pp. 1–22, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
M. Moakher, “A differential geometric approach to the geometric mean of symmetric positive-definite matrices,” SIAM Journal on Matrix Analysis and Applications, vol. 26, no. 3, pp. 735–747, 2005.
View at: Publisher Site | Google Scholar | MathSciNet

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Copyright © 2015 Pattrawut Chansangiam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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