Chapter Five

---

---

Positive Definite Functions

Positive definite functions arise naturally in many areas of mathematics. In this chapter we study some of their basic properties, construct some examples, and use them to derive interesting results about positive matrices.

5.1  BASIC PROPERTIES

Positive definite sequences were introduced in Section 1.1.3. We repeat the definition. A (doubly infinite) sequence of complex numbers {a_n : n ∈ } is said to be positive definite if for every positive integer N, we have

for every finite sequence of complex numbers ξ₀, ξ₁, . . . , ξ_N−1. This condition is equivalent to the requirement that for each N = 1, 2, . . . , the N × N matrix

is positive.

From this condition it is clear that we must have

A complex-valued function φ on is said to be positive definite if for every positive integer N, we have

for every choice of real numbers x₀, x₁, . . . , x_N−1, and complex numbers ξ₀, ξ₁, . . . , ξ_N−1. In other words φ is positive definite if for each N = 1, 2, . . . the N × N matrix

is positive for every choice of real numbers x₀, . . . , x_N−1. It follows from this condition that

Thus every positive definite function is bounded, and its maximum absolute value is attained at 0.

5.1.1  Exercise

Let φ(x) be the characteristic function of the set ; i.e., φ(x) = 1 if and φ(x) = 0 if . Show that φ is positive definite. This remains true when is replaced by any additive subgroup of .

5.1.2  Lemma

If φ is positive definite, then for all x₁, x₂

Proof.   Assume, without loss of generality, that φ(0) = 1. Choose x₀ = 0. The 3 × 3 matrix

is positive. So for every vector u, the inner product . Choose u = (z, 1, −1) where z is any complex number. This gives the inequality

Now choose . This gives

Exercise 5.1.1 showed us that a positive definite function φ need not be continuous. Lemma 5.1.2 shows that if the real part of φ is continuous at 0, then φ is continuous everywhere on .

5.1.3  Exercise

Let φ(x) be positive definite. Then

(i)  is positive definite.

(ii)  For every real number t the function φ(tx) is positive definite.

5.1.4  Exercise

(i)   If φ₁, φ₂ are positive definite, then so is their product φ₁φ₂. (Schur’s theorem.)

(ii)  If φ is positive definite, then |φ|² is positive definite. So is Re φ.

5.1.5  Exercise

(i)   If φ₁, . . . , φ_n are positive definite, and a₁, . . . , a_n are positive real numbers, then a₁φ₁ + · · · + a_nφ_n is positive definite.

(ii)  If {φ_n} is a sequence of positive definite functions and φ_n(x) → φ(x) for all x, then φ is positive definite.

(iii)  If φ is positive definite, then e^ϕ is positive definite, and so is e^ϕ+a for every .

(iv) If φ(x) is a measurable positive definite function and f(t) is a nonnegative integrable function, then is positive definite.

(v)  If µ is a finite positive Borel measure on and φ(x) a measurable positive definite function, then the function is positive definite. (The statement (iv) is a special case of (v).)

Let I be any interval and let K(x, y) be a bounded continuous complex-valued function on I × I. We say K is a positive definite kernel if

for every continuous integrable function f on the interval I.

5.1.6  Exercise

(i)   A bounded continuous function K(x, y) on I × I is a positive definite kernel if and only if for all choices of points x₁, . . . , x_N in I, the N × N matrix [[K(x_i, x_j)]] is positive.

(ii)  A bounded continuous function φ on is positive definite if and only if the kernel K(x, y) = φ(x − y) is positive definite.

5.2  EXAMPLES

5.2.1

The function φ(x) = e^ix is positive definite since

Exercise: Write the matrix (5.5) in this case as uu^∗ where u is a column vector.

This example is fundamental. It is a remarkable fact that all positive definite functions can be built from this one function by procedures outlined in Section 5.1.

5.2.2

The function φ(x) = cos x is positive definite.

Exercise: sin x is not positive definite. The matrix A in Exercise 1.6.6 has entries a_ij = cos(x_i − x_j) where x_i = 0, π/4, π/2, 3π/4. It follows that | cos x| is not positive definite.

5.2.3

For each , φ(x) = e^itx is a positive definite function.

5.2.4

Let and let f(t) ≥ 0. Then

is positive definite. More generally, if µ is a positive finite Borel measure on , then

is positive definite. The function is called the Fourier transform of f and is called the Fourier-Stieltjes transform of µ. Both of them are bounded uniformly continuous functions.

These transforms give us many interesting positive definite functions.

5.2.5

One of the first calculations in probability theory is the computation of an integral:

The integral on the right hand side can be evaluated using Cauchy’s theorem. Let C be the rectangular contour with vertices −R, R, R + ix, −R + ix. The integral of the analytic function f(z) = e^−z2/2 along this contour is zero. As R → ∞, the integral along the two vertical sides of this contour goes to zero. Hence

So,

(This shows that, with a suitable normalization, the function e^−x2/2 is its own Fourier transform.) Thus for each a ≥ 0, the function φ(x) = e^−ax2 is positive definite.

5.2.6

The function φ(x) = sin x/x is positive definite. To see this one can use the product formula

and observe that each of the factors in this infinite product is a positive definite function. Alternately, we can use the formula.

(This integral is the Fourier transform of the characteristic function .)

We have tacitly assumed here that φ(0) = 1. This is natural. If we assign φ(0) any value larger than 1, the resulting (discontinuous) function is also positive definite.

5.2.7

The integral

shows that the function φ(x) = 1/(1 + ix) is positive definite. The functions 1/(1 − ix) and 1/(1 + x²) are positive definite.

5.2.8

The integral formulas

and

show that the functions 1/(1 + x²) and e^−|x| are positive definite. (They are nonnegative and are, up to a constant factor, Fourier transforms of each other.)

5.2.9

From the product representations

and

one sees (using the fact that 1/(1 + a²x²) is positive definite) that the functions x/(sinh x) and 1/(cosh x) are positive definite. (Contrast this with 5.2.6 and 5.2.2.)

5.2.10

For 0 < α < 1, we have from (5.13)

Each factor in this product is of the form

This shows that the function sinh αx/sinh x is positive definite for 0 ≤ α ≤ 1. In the same way using (5.14) one can see that cosh αx/cosh x is positive definite for −1 ≤ α ≤ 1. The function

is positive definite for −1/2 ≤ α ≤ 1/2, as it is the product of two such functions.

5.2.11

The integral

shows that tanh x/x is a positive definite function.

(Once again, it is natural to assign the functions sinh x/x, sinh αx/x and tanh x/x the values 1, α and 1, respectively, at x = 0. Then the functions under consideration are continuous and positive definite. Assigning them larger values at 0 destroys continuity but not positive definiteness.)

5.2.12

One more way of constructing positive definite functions is by convolutions of functions in . For any function f let be the function defined as . If then the function defined as

is a continuous function vanishing at ∞. It is a positive definite function since

5.2.13

Let R be a positive real number. The tent function (with base R) is defined as

A calculation shows that

So, ∆_R(x) is positive definite for all R > 0.

5.2.14

For R > 0, let δ_R be the continuous function defined as

From 5.2.6 it follows that δ_R is positive definite. The family {δ_R}_R>0 is called the Fejér kernel on R. It has the following properties (required of any “summability kernel” in Fourier analysis):

(i)   δ_R(x) ≥ 0 for all x, and for all R > 0.

(ii)  For every a > 0, δ_R(x) → 0 uniformly outside [−a, a] as R → ∞.

(iii)  for every a > 0.

(iv)  for all R > 0.

Property (iv) may be proved by contour integration, for example.

The functions ∆_R and δ_R are Fourier transforms of each other (up to constant factors). So the positive definiteness of one follows from the nonnegativity of the other.

5.2.15

In this section we consider functions like the tent functions of Section 5.2.13.

a  Let φ be any continuous, nonnegative, even function. Suppose φ(x) = 0 for |x| ≥ R, and φ is convex and monotonically decreasing in the interval [0, R). Then φ is a uniform limit of sums of the form , where a_k ≥ 0 and R_k ≤ R. It follows from 5.2.13 that φ is positive definite.

b  The condition that φ is supported in [−R, R] can be dropped. Let φ be any continuous, nonnegative, even function that is convex and monotonically decreasing in [0, ∞). Let . Then φ is a uniform limit of sums of the form , where a_k ≥ 0 and R_k > 0. Hence φ is positive definite. This is Pólya’s Theorem.

c  Let φ be any function satisfying the conditions in Part a of this section, and extend it to all of as a periodic function with period 2R. Since φ is even, the Fourier expansion of φ does not contain any sin terms. It can be seen from the convexity of φ in (0, R) that the coefficients a_n in the Fourier expansion

are nonnegative. Hence φ is positive definite.

5.2.16

Using 5.2.15 one can see that the following functions are positive definite:

The special case a = 1 of (iii) was seen in 5.2.8. The next theorem provides a further extension.

5.2.17  Theorem

The function φ(x) = exp(−|x|^a) is positive definite for 0 ≤ a ≤ 2.

Proof.   Let 0 < a < 2. A calculation shows that

where

(The assumption a < 2 is needed to ensure that this last integral is convergent. At 0 the numerator in the integrand has a zero of order 2.) Thus we have

where dµ(t) = C_a dt. Let φ_n be defined as

The second integral in the last line is a number, say c_n, while the first is a function, say g_n(x). This function is positive definite since cos xt is positive definite for all t. So, for each n, the function exp φ_n(x) is positive definite by Exercise 5.1.5 (iii). Since lim_n→∞ exp φ_n(x) = exp(−|x|^a), this function too is positive definite for 0 < a < 2. Again, by continuity, this is true for a = 2 as well.     ■

For a > 2 the function exp(−|x|^a) is not positive definite. This is shown in Exercise 5.5.8.

5.2.18

The equality

shows that the matrix on the left-hand side is positive. Thus the n × n matrix A with entries a_ij = min(i, j) is positive. This can be used to see that the kernel K(x, y) = min(x, y) is positive definite on [0, ∞) × [0, ∞).

5.2.19  Exercise

Let B be the n × n matrix with entries b_ij = 1/max(i, j). Show that this matrix is positive by an argument similar to the one in 5.2.18.

Note that if A is the matrix in 5.2.18, then B = DAD, where D = diag(1, 1/2, . . . , 1/n).

5.2.20  Exercise

Let λ₁, . . . , λ_n be positive real numbers. Let A and B be the n × n matrices whose entries are a_ij = min(λ_i, λ_j) and b_ij = 1/ max(λ_i, λ_j), respectively. Show that A and B are positive definite.

5.2.21  Exercise

Show that the matrices A and B defined in Exercise 5.2.20 are infinitely divisible.

5.2.22  Exercise

Let 0 < λ₁ ≤ λ₂ ≤ · · · ≤ λ_n and let A be the symmetric matrix whose entries a_ij are defined as a_ij = λ_i/λ_j for 1 ≤ i ≤ j ≤ n. Show that A is infinitely divisible.

5.2.23  Exercise

Let λ₁, λ₂, . . . , λ_n be real numbers. Show that the matrix A whose entries are

is infinitely divisible. [Hint: Use Pólya’s Theorem.]

5.3  LOEWNER MATRICES

In this section we resume, and expand upon, our discussion of operator monotone functions. Recall some of the notions introduced at the end of Chapter 2. Let C¹(I) be the space of continuously differentiable real-valued functions on an open interval I. The first divided difference of a function f in C¹(I) is the function f^[1] defined on I × I as

Let be the collection of all n × n Hermitian matrices whose eigenvalues are in I. This is an open subset in the real vector space consisting of all Hermitian matrices. The function f induces a map from into .

If f ∈ C¹(I) and we define f^[1](A) as the matrix whose i, j entry is f^[1](λ_i, λ_j), where λ₁, . . . , λ_n are the eigenvalues of A. This is called the Loewner matrix of f at A.

The function f on is differentiable. Its derivative at A, denoted as Df(A), is a linear map on characterized by the condition

for all . We have

An interesting expression for this derivative in terms of Loewner matrices is given in the following theorem.

5.3.1  Theorem

Let f ∈ C¹(I) and . Then

where ◦ denotes the Schur product in a basis in which A is diagonal.

One proof of this theorem can be found in MA (Theorem V.3.3). Here we give another proof based on different ideas.

Let [X, Y ] stand for the Lie bracket: [X, Y ] = XY − Y X. If X is Hermitian and Y skew-Hermitian, then [X, Y ] is Hermitian.

5.3.2  Theorem

Let f ∈ C¹(I) and . Then for every skew-Hermitian matrix K

Proof.   The exponential e^tK is a unitary matrix for all . From the series representation of e^tK one can see that

Since f is in the class C¹, this is equal to

For each , the collection

is a linear subspace of . On we have an inner product = tr XY . With respect to this inner product, the orthogonal complement of C_A is the space

(It is easy to prove this. If H commutes with A, then

Proof of Theorem 5.3.1.    Choose an orthonormal basis in which A = diag(λ₁, . . . , λ_n). Let ; i.e., H = [A, K] for some skew-Hermitian matrix K. By (5.21), Df(A)(H) = [f(A), K]. The entries of this matrix are

These are the entries of f^[1](A) ◦ H also. Thus the two sides of (5.20) are equal when . Now let H belong to the complementary space . The theorem will be proved if we show that the equality (5.20) holds in this case too. But this is easy. Since H commutes with A, we may assume H too is diagonal, H = diag(h₁, . . . , h_n). In this case the two sides of (5.20) are equal to the diagonal matrix with entries f^′(λ_i)h_i on the diagonal.     ■

The next theorem says that f is operator monotone on I if and only if for all n and for all the Loewner matrices f^[1](A) are positive. (This is a striking analogue of the statement that a real function f is monotonically increasing if and only if f^′(t) ≥ 0.)

5.3.3  Theorem

Let f ∈ C¹(I). Then f is operator monotone on I if and only if f^[1](A) is positive for every Hermitian matrix A whose eigenvalues are contained in I.

Proof.   Suppose f is operator monotone. Let and let H be the positive matrix with all its entries equal to 1. For small positive t, A + tH is in . We have A + tH ≥ A, and hence f(A + tH) ≥ f(A). This implies Df(A)(H) ≥ O. For this H, the right-hand side of (5.20) is just f^[1](A), and we have shown this is positive.

To prove the converse, let A₀, A₁ be matrices in with A₁ ≥ A₀. Let A(t) = (1 − t)A₀ + tA₁, 0 ≤ t ≤ 1. Then A(t) is in . Our hypothesis says that f^[1](A(t)) is positive. The derivative A^′(t) = A₁ − A₀ is positive, and hence the Schur product f^[1](A(t)) ◦ A^′(t) is positive. By Theorem 5.3.1 this product is equal to Df(A(t))(A^′(t)). Since

and the integrand is positive for all t, we have f(A₁) ≥ f(A₀).     ■

We have seen some examples of operator monotone functions in Section 4.2. Theorem 5.3.3 provides a direct way of proving operator monotonicity of these and other functions. The positivity of the Loewner matrices f^[1](A) is proved by associating with them some positive definite functions. Some examples follow.

5.3.4

The function

is operator monotone on any interval I that does not contain the point −d/c.

To see this write down the Loewner matrix f^[1](A) for any A ∈ . If λ₁, . . . , λ_n are the eigenvalues of A, this Loewner matrix has entries

This matrix is congruent to the matrix with all entries 1, and is therefore positive.

5.3.5

The function f(t) = t^r is operator monotone on (0, ∞) for 0 ≤ r ≤ 1.

A Loewner matrix for this function is a matrix V with entries

The numbers λ_i are positive and can, therefore, be written as e^x_i for some x_i. We have then

This matrix is congruent to the matrix with entries

Since sinh rx/(sinh x) is a positive definite function for 0 ≤ r ≤ 1 (see 5.2.10), this matrix is positive.

5.3.6  Exercise

The function f(t) = t^r is not operator monotone on (0, ∞) for any real number r outside [0, 1].

5.3.7

The function f(t) = log t is operator monotone on (0, ∞).

A Loewner matrix in this case has entries

The substitution λ_i = e^x_i reduces this to

This matrix is positive since the function x/(sinh x) is positive definite. (See 5.2.9.)

Another proof of this is obtained from the equality

For each t the matrix [[1/(λ_i + t)(λ_j + t)]] is positive. (One more proof of operator monotonicity of the log function was given in Exercise 4.2.5.)

5.3.8

The function f(t) = tan t is operator monotone on .

In this case a Loewner matrix has entries

This matrix is positive since the function sin x/x is positive definite. (See 5.2.6.)

5.3.9  Exercise

For 0 ≤ r ≤ 1 let f be the map f(A) = A^r on the space of positive definite matrices. Show that

5.4  NORM INEQUALITIES FOR MEANS

The theme of this section is inequalities for norms of some expressions involving positive matrices. In the case of numbers they reduce to some of the most fundamental inequalities of analysis.

As a prototype consider the arithmetic-geometric mean inequality for positive numbers a, b. There are many different directions in which one could look for a generalization of this to positive matrices A, B. One version that involves the somewhat subtle concept of a matrix geometric mean is given in Section 4.1. Instead of matrices we could compare numbers associated with them. Thus, for example, we may ask whether

for every unitarily invariant norm. This is indeed true. There is a more general version of this inequality that is easier to prove: we have

for every X. What makes it easier is a lovely trick. It is enough to prove (5.24) in the special case A = B. (The inequality (5.23) is a vacuous statement in this case.) Suppose we have proved

for all matrices X and positive A. Then given X and positive A, B we may replace A and X in (5.25) by the 2 × 2 block matrices and . This gives the inequality (5.24).

Since the norms involved are unitarily invariant we may assume that A is diagonal, A = diag(λ₁, . . . , λ_n). Then we have

where Y is the matrix with entries

This matrix is congruent to the Cauchy matrix—the one whose entries are 1/(λ_i + λ_j). Since that matrix is positive (Exercise 1.1.2) so is Y . All the diagonal entries of Y are equal to 1. So, using Exercise 2.7.12 we get the inequality (5.25) from (5.26).

The inequalities that follow are proved using similar arguments. Matrices that occur in the place of (5.27) are more complicated and their positivity is not as easy to establish. But in Section 5.2 we have done most of the work that is needed.

5.4.1  Theorem

Let A, B be positive and let X be any matrix. Then for 0 ≤ ν ≤ 1 we have

Proof.   Follow the arguments used above in proving (5.24). To prove the second inequality in (5.28) we need to prove that the matrix Y whose entries are

is positive for 0 < ν < 1. (When ν = 1/2 this reduces to (5.27).) Writing

we see that Y is congruent to the matrix Z whose entries are

This matrix is like the one in 5.3.5. The argument used there reduces the question of positivity of Z to that of positive definiteness of the function cosh αx/(cosh x) for −1 < α < 1. In 5.2.10 we have seen that this function is indeed positive definite. The proof of the first inequality in (5.28) is very similar to this, and is left to the reader.     ■

5.4.2  Exercise

Show that for 0 ≤ ν ≤ 1

5.4.3  Exercise

For the Hilbert-Schmidt norm we have

for positive matrices A and 0 < ν < 1. This is not always true for the operator norm || · ||.

5.4.4  Exercise

For any matrix Z let

Let A be a positive matrix and let X be a Hermitian matrix. Let S = A^νXA^1−ν, T = νAX + (1 − ν)XA. Show that for 0 ≤ ν ≤ 1

In Chapter 4 we defined the logarithmic mean of a and b. This is the quantity

A proof of the inequality (4.3) using the ideas of Section 5.3 is given below.

5.4.5  Lemma

Proof.   Put a = e^x and b = e^y. A small calculation reduces the job of proving the first inequality in (5.33) to showing that t ≤ sinh t for t > 0, and the second to showing that tanh t ≤ t for all t > 0. Both these inequalities can be proved very easily.     ■

5.4.6  Exercise

Show that for A, B positive and for every X

This matrix version of the arithmetic-logarithmic-geometric mean inequality can be generalized to all unitarily invariant norms.

5.4.7  Theorem

For every unitarily invariant norm we have

Proof.   The idea of the proof is very similar to that of Theorem 5.4.1. Assume B = A, and suppose A is diagonal with entries λ₁, . . . , λ_n on the diagonal. The matrix A^1/2XA^1/2 is obtained from by entrywise multiplication with the matrix Y whose entries are

This matrix is congruent to one with entries

We have seen in 5.3.7 that this matrix is positive. That proves the first inequality in (5.35).

The matrix is the Schur product of with the matrix W whose entries are

Making the substitution λ_i = e^x_i, we have

This matrix is positive since the function tanh x/x is positive definite. (See 5.2.11.) That proves the second inequality in (5.35).     ■

5.4.8  Exercise

A refinement of the inequalities (5.28) and (5.35) is provided by the assertion

for 1/4 ≤ ν ≤ 3/4. Prove this using the fact that (x cosh αx)/ sinh x is a positive definite function for −1/2 ≤ α ≤ 1/2. (See 5.2.10.)

5.4.9  Exercise

Let H, K be Hermitian, and let X be any matrix. Show that

This is a matrix version of the inequality |sin x| ≤ |x|.

5.4.10  Exercise

Let H, K and X be as above. Show that

5.4.11  Exercise

Let A, B be positive matrices. Show that

Hence, if H, K are Hermitian, then

for every matrix X.

5.5  THEOREMS OF HERGLOTZ AND BOCHNER

These two theorems give complete characterizations of positive definite sequences and positive definite functions, respectively. They have important applications throughout analysis. For the sake of completeness we include proofs of these theorems here. Some basic facts from functional analysis and Fourier analysis are needed for the proofs. The reader is briefly reminded of these facts.

Let M[0, 1] be the space of complex finite Borel measures on the interval [0, 1]. This is equipped with a norm , and is the Banach space dual of the space C[0, 1]. If converges to for every f ∈ C[0, 1], we say that the sequence {µ_n} in M[0, 1] converges to µ in the weak^∗ topology.

A basic fact about this convergence is the following theorem called Helly’s Selection Principle.

5.5.1  Theorem

Let {µ_n} be a sequence of probability measures on [0, 1]. Then there exists a probability measure µ and a subsequence {µ_m} of {µ_n} such that µ_m converges in the weak^∗ topology to µ.

Proof.   The space C[0, 1] is a separable Banach space. Choose a sequence {f_j} in C[0, 1] that includes the function 1 and whose linear combinations are dense in C[0, 1]. Since ||µ_n|| = 1, for each j we have for all n. Thus for each j is a bounded sequence of positive numbers. By the diagonal procedure, we can extract a subsequence {µ_m} such that for each j, the sequence converges to a limit, say ξ_j, as m → ∞.

If is any (finite) linear combination of the f_j, let

This is a linear functional on the linear span of {f_j}, and |Λ₀(f)| ≤ ||f|| for every f in this span. By continuity Λ₀ has an extension Λ to C[0, 1] that satisfies |Λ(f)| ≤ ||f|| for all f in C[0, 1]. This linear functional Λ is positive and unital. So, by the Riesz Representation Theorem, there exists a probability measure µ on [0, 1] such that for all f ∈ C[0, 1].

Finally, we know that converges to for every f in the span of {f_j}. Since such f are dense and the µ_m are uniformly bounded, this convergence persists for every f in C[0, 1].     ■

Theorem 5.5.1 is also a corollary of the Banach Alaoglu theorem. This says that the closed unit ball in the dual space of a Banach space is compact in the weak^∗ topology. If a Banach space X is separable, then the weak^∗ topology on the closed unit ball of its dual X^∗ is metrizable.

5.5.2  Herglotz’ Theorem

Let be a positive definite sequence and suppose a₀ = 1. Then there exists a probability measure µ on [−π, π] such that

Proof.   The positive definiteness of {a_n} implies that for every real x we have

This inequality can be expressed in another form

Let f_N(x) be the function given by the last sum. Then

For any Borel set E in [−π, π], let

Then µ_N is a probability measure on [−π, π]. Apply Helly’s selection principle to the sequence {µ_N }. There exists a probability measure µ to which (a subsequence of ) µ_N converges in the weak^∗ topology. Thus for every n

We remark that the sum

is called the Fejér kernel and is much used in the study of Fourier series.

The condition a₀ = 1 in the statement of Herglotz’ theorem is an inessential normalization. This can be dropped; then µ is a finite positive measure with ||µ|| = a₀.

Bochner’s theorem, in the same spirit as Herglotz’, says that every continuous positive definite function on is the Fourier-Stieltjes transform of a finite positive measure on R. The proof needs some approximation arguments. For the convenience of the reader let us recall some basic facts.

For we write for its Fourier transform defined as

This function is in , the class of continuous functions vanishing at ∞. We write

for the inverse Fourier transform of f. If the function is in (and this is not always the case) then

The Fourier transform on the space is defined as follows. Let . Then is defined as above. One can see that and the map is an L₂-isometry on the space . This space is dense in . So the isometry defined on it has a unique extension to all of . This unitary operator on is denoted again by . The inverse of the map is defined by inverting this unitary operator. The fact that the Fourier transform is a bijective map of onto itself makes some operations in this space simpler.

Let δ_R be the function defined in 5.2.14. The family {δ_N} is an approximate identity: as N → ∞, the convolution δ_N ∗ g converges to g in an appropriate sense. The “appropriate sense” for us is the following.

If g is either an element of , or a bounded measurable function, then

In the discussion that follows we ignore constant factors involving 2π. These do not affect our conclusions in any way.

The Fourier transform “converts convolution into multiplication;” i.e.,

The Riesz representation theorem and Helly’s selection principle have generalizations to the real line. The space is a separable Banach space. Its dual is the space of finite Borel measures on with norm . Every bounded sequence {µ_n} in has a weak^∗ convergent subsequence {µ_m}; i.e., for every converges to . This too is a special case of the Banach-Alaoglu theorem.

5.5.3  Bochner’s Theorem

Let φ be any function on the real line that is positive definite and continuous at 0. Then there exists a finite positive measure µ such that

Proof.   By Lemma 5.1.2, φ is continuous everywhere. Suppose in addition that . Using (5.6) we see that

Thus φ is in the space also. Hence, there exists such that

Let ∆_N(x) be the tent function defined in (5.16). Then

This integral (of a continuous function over a bounded interval) is a limit of Riemann sums. Let x_j = j N/K, −K ≤ j ≤ K. The last integral is the limit, as K → ∞, of sums

These sums can be expressed in another way:

where c(K, N) is a positive number. (See the proof of Herglotz’ theorem where two sums of this type were involved.) Since φ is positive definite, the sum in (5.41) is nonnegative. Hence, the integral (5.41), being the limit of such sums, is nonnegative. As N → ∞ the integral in (5.40) tends to the one in (5.39). So, that integral is nonnegative too. Thus f(t) ≥ 0.

Now let φ be any continuous positive definite function and let

Since φ is bounded, φ_n is integrable. Since φ(x) and e^−x2/n are positive definite, so is their product φ_n(x). Thus by what we have proved in the preceding paragraph, for each n

We have the relation δ_N ∗ φ_n = (∆_Nf_n)^{^}, i.e.,

At x = 0 this gives

Let N → ∞. This shows

i.e., and ||f_n||₁ ≤ φ(0). Let dµ_n(t) = f_n(t)dt. Then {µ_n} are positive measures on and ||µ_n|| ≤ φ(0). So, by Helly’s selection principle, there exists a positive measure µ, with ||µ|| ≤ φ(0), to which (a subsequence of ) µ_n converges in the weak^∗ topology.

The equation (5.42) says

Keep N fixed and let n → ∞. For the right-hand side of (5.43) use the weak^∗ convergence of µ_n to µ, and for the left-hand side the Lebesgue-dominated convergence theorem. This gives

Now let N → ∞. Since φ is a bounded measurable function, by (5.37) the left-hand side of (5.44) goes to φ(x) a.e. The right-hand side converges by the bounded convergence theorem. This shows

Since the two sides are continuous functions of x, this equality holds everywhere.     ■

Of the several examples of positive definite functions in Section 5.2 some were shown to be Fourier transforms of nonnegative integrable functions. (See 5.2.5 - 5.2.8.) One can do this for some of the other functions too.

5.5.4

The list below gives some functions φ and their Fourier transforms φ (ignoring constant factors).

Let φ be a continuous positive definite function. Then the measure µ associated with φ via the formula (5.38) is a probability measure if and only if φ(0) = 1. In this case φ is called a characteristic function.

5.5.5  Proposition

Let φ be a characteristic function. Then

for all x and n = 1, 2, . . . .

Proof.   By elementary trigonometry

An iteration leads to the inequality

From (5.38) we have

5.5.6  Corollary

Suppose φ is a positive definite function and φ(x) = φ(0) + o(x²); i.e.,

Then φ is a constant.

Proof.   We may assume that φ(0) = 1. Then using the Proposition above we have for all x and n

The hypothesis on φ implies that the last expression goes to zero as n → ∞. Hence, Re φ(x) = 1 for all x. But then φ(x) ≡ 1.     ■

5.5.7  Exercise

Suppose φ is a characteristic function, and φ(x) = 1 + o(x) + o(x²) in a neighbourhood of 0, where o(x) is an odd function. Then φ ≡ 1. [Hint: consider φ(x)φ(−x).]

5.5.8  Exercise

The functions e^−x4 , 1/(1 + x⁴), and e^−|x|a for a > 2, are not positive definite.

Bochner’s theorem can be used also to show that a certain function is not positive definite by showing that its Fourier transform is not everywhere nonnegative.

5.5.9  Exercise

Use the method of residues to show that for all t > 0

It follows from Bochner’s theorem that the function f(x) = 1/(1 + x⁴) is not positive definite.

5.6  SUPPLEMENTARY RESULTS AND EXERCISES

5.6.1  Exercise

Let U be a unitary operator on any separable Hilbert space . Show that for each unit vector x in the sequence

is positive definite.

This observation is the first step on one of the several routes to the spectral theorem for operators in Hilbert space. We indicate this briefly.

Let U be a unitary operator on . By Exercise 5.6.1 and Herglotz’ theorem, for each unit vector x in , there exists a probability measure µ_x on the interval [−π, π] such that

Using a standard technique called polarisation, one can obtain from this, for each pair x, y of unit vectors a complex measure µ_x,y such that

Now for each Borel subset E ⊂ [−π, π] let P(E) be the operator on defined by the relation

It can be seen that P(E) is an orthogonal projection and that P(·) is countably additive on the Borel σ-algebra of [−π, π]. In other words P(·) is a projection-valued measure. We can then express U as an integral

This is the spectral theorem for unitary operators. The spectral theorem for self-adjoint operators can be obtained from this using the Cayley transform.

5.6.2  Exercise

Let B be an n × n Hermitian matrix. Show that for each unit vector u the function

is a positive definite function on . Use this to show that the functions tr e^itB and det e^itB are positive definite.

5.6.3  Exercise

Let A, B be n × n Hermitian matrices and let

Is φ a positive definite function? Show that this is so if A and B commute.

The general case of the question raised above is a long-standing open problem in quantum statistical mechanics. The Bessis-Moussa-Villani conjecture says that the function φ in (5.50) is positive definite for all Hermitian matrices A and B.

The purpose of the next three exercises is to calculate Fourier transforms of some functions that arose in our discussion.

5.6.4  Exercise

Let φ(x) = 1/cosh x. Its Fourier transform is

This integral may be evaluated by the method of residues. Let f be the function

Then

For any R > 0 the rectangular contour with vertices −R, R, R+iπ and −R + iπ contains one simple pole, z = iπ/2, of f inside it. Integrate f along this contour and then let |R| → ∞. The contribution of the two vertical sides goes to zero. So

where Res_z=z0 f(z) is the residue of f at a pole z₀.

A calculation shows that

5.6.5  Exercise

More generally consider the function

Integrate the function

along the rectangular contour with vertices −R, R, R + i2π and −R + i2π. The function f has two simple poles z = i(π ± arccos a) inside this rectangle. Proceed as in Exercise 5.6.4 to show

It is plain that . Hence by Bochner’s theorem φ(x) is positive definite for −1 < a < 1. By a continuity argument it is positive definite for a = 1 as well.

5.6.6  Exercise

Now consider the function

Use the function f and the rectangular contour of Exercise 5.6.5. Now f has two simple poles z = ± arccosh t+iπ inside this rectangle. Show that

It is plain that is negative for some values of t. So the function φ in (5.53) is not positive definite for any a > 1.

5.6.7  Exercise

Let λ₁, . . . , λ_n be positive numbers and let Z be the n× n matrix with entries

Show that if −2 < t ≤ 2, then Z is positive definite; and if t > 2 then there exists an n > 2 for which this matrix is not positive definite. (See Exercise 1.6.4.)

5.6.8  Exercise

For 0 < a < 1, let f_a be the piecewise linear function defined as

Show that f_a is not positive definite. Compare this with 5.2.13 and 5.2.15. Express f_a as the convolution of two characteristic functions.

The technique introduced in Section 4 is a source of several interesting inequalities. The next two exercises illustrate this further.

5.6.9  Exercise

(i)  Let A be a Hermitian matrix. Use the positive definiteness of the function sech x to show that for every matrix X

(ii)  Now let A be any matrix. Apply the result of (i) to the matrices and show that

for every matrix X. Replacing A by iA, one gets

5.6.10  Exercise

Let A, B be normal matrices with ||A|| ≤ 1 and ||B|| ≤ 1. Show that for every X we have

The inequalities proved in Section 5.4 have a leitmotiv. Let M(a, b) be any mean of positive numbers a and b satisfying the conditions laid down at the beginning of Chapter 4. Let A be a positive definite matrix with eigenvalues λ₁ ≥ · · · ≥ λ_n. Let M(A, A) be the matrix with entries

Many of the inequalites in Section 5.4 say that for certain means M₁ and M₂

for all X. We have proved such inequalites by showing that the matrix Y with entries

is positive definite. This condition is also necessary for (5.55) to be true for all X.

5.6.11  Proposition

Let M₁(a, b) and M₂(a, b) be two means. Then the inequality (5.55) is true for all X if and only if the matrix Y defined by (5.56) is positive definite.

Proof.   The Schur product by Y is a linear map on . The inequality (5.55) says that this linear map on the space equipped with the norm || · || is contractive. Hence it is contractive also with respect to the dual norm || · ||₁; i.e.,

Choose X to be the matrix with all entries equal to 1. This gives ||Y ||₁ ≤ n. Since Y is Hermitian

where λ_i(Y ) are the eigenvalues of Y. Since y_ii = 1 for all i, we have

Thus . But this is possible only if λ_i(Y ) ≥ 0 for all i. In other words Y is positive.     ■

Let us say that M₁ ≤ M₂ if M₁(a, b) ≤ M₂(a, b) for all positive numbers a and b; and M₁ << M₂ if for every n and every choice of n positive numbers λ₁, . . . , λ_n the matrix (5.56) is positive definite. If M₁ << M₂ the inequality (5.55) is true for all positive matrices A and all matrices X. Clearly M₁ ≤ M₂ if M₁ << M₂. The converse is not always true.

5.6.12  Exercise

Let A(a, b) and G(a, b) be the arithmetic and the geometric means of a and b. For 0 ≤ α ≤ 1 let

Clearly we have F_α ≤ F_β whenever α ≤ β. Use Exercise 5.6.6 to show that F_α << F_β if and only if .

Using Exercise 2.7.12 one can see that if M₁ << M₂, then the inequality (5.55) is true for all unitarily invariant norms. The weaker condition M₁ ≤ M₂ gives this inequality only for the Hilbert-Schmidt norm || · ||₂.

In Exercises 1.6.3, 1.6.4, 5.2.21, 5.2.22 and 5.2.23 we have outlined simple proofs of the infinite divisibility of some special matrices. These proofs rely on arguments specifically tailored to suit the matrices at hand. In the next few exercises we sketch proofs of some general theorems that are useful in this context.

An n × n Hermitian matrix A is said to be conditionally positive definite if for all such that x₁ + · · · + x_n = 0. (The term almost positive definite is also used sometimes.)

5.6.13  Proposition

Let A = [[a_ij]] be an n × n conditionally positive definite matrix. Then there exist a positive definite matrix B = [[b_ij]] and a vector y = (y₁, . . . , y_n) in such that

Proof.   Let J be the n × n matrix all of whose entries are equal to 1/n. For any vector let x^# = Jx and . Since we have

Let B = A − AJ − JA + JAJ. The inequality above says that

In other words B is positive definite.

If C₁, . . . , C_n are the columns of the matrix A, then the jth column of the matrix JA has all its entries equal to , the number obtained by averaging the entries of the column C_j. Likewise, if R₁, . . . , R_n are the rows of A, then the ith row of AJ has all its entries equal to . Since A is Hermitian is the complex conjugate of . The matrix JAJ has all its entries equal to . Thus the i, j entry of the matrix AJ + JA − JAJ is equal to . Let y be the vector

Then the equation (5.57) is satisfied.     ■

5.6.14  Exercise

Let A = [[a_ij]] be a conditionally positive definite matrix. Show that the matrix [[e^a_ij ]] is positive definite. [Hint: If B = [[b_ij]] is positive definite, then [[e^b_ij ]] is positive definite. Use Proposition 5.6.13.]

5.6.15  Exercise

Let A = [[a_ij]] be an n × n matrix with positive entries and let L = [[log a_ij]]. Let E be the matrix all whose entries are equal to 1. Note that Ex = 0 if x₁ + · · · + x_n = 0.

(i)  Suppose A is infinitely divisible. Let x be any vector with x₁ + · · · + x_n = 0, and for r > 0 let A^(r) be the matrix with entries a^r_ij. Then

Let r ↓ 0. This gives

Thus L is conditionally positive definite.

(ii)  Conversely, if L is conditionally positive definite, then so is rL for every r ≥ 0. Use Exercise 5.6.14 to show that this implies A is infinitely divisible.

Thus a Hermitian matrix A with entries a_ij > 0 is infinitely divisible if and only if the matrix L = [[log a_ij]] is conditionally positive definite. The next exercise gives a criterion for conditional positive definiteness.

5.6.16  Exercise

Given an n×n Hermitian matrix B = [[b_ij]] let D be the (n−1)×(n−1) matrix with entries

Show that B is conditionally positive definite if and only if D is positive definite.

We now show how the results of Exercises 5.6.14–5.6.16 may be used to prove the infinite divisibility of an interesting matrix.

For any n, the n × n Pascal matrix A is the matrix with entries

The entries of the Pascal triangle occupy the antidiagonals of A. Thus the 4 × 4 Pascal matrix is

5.6.17  Exercise

Prove the combinatorial identity

[Hint: Separate i + j objects into two groups, the first containing i objects and the second j objects. If we choose i − k objects from the first group and k from the second, we have chosen i objects out of i + j.]

5.6.18  Exercise

Show that

Use this to conclude that the Pascal matrix is a Gram matrix and is thus positive definite.

5.6.19  Exercise

Let L be the n × n lower triangular matrix whose rows are the rows of the Pascal triangle. Thus for n = 4

Show that A = LL^∗. This gives another proof of the positive definiteness of the Pascal matrix A.

5.6.20  Exercise

For every n, the n × n Pascal matrix is infinitely divisible. Prove this statement following the steps given below.

(i)  Use the results of Exercises 5.6.15 and 5.6.16. If B has entries b_ij = log a_ij, where a_ij are defined by (5.59), then the entries d_ij defined by (5.58) are given by

We have to show that the matrix D = [[d_ij]] is positive definite.

(ii)  For x > 0 we have

where µ is the probability measure on [0, ∞) defined as dµ(t) = dt/t². Use this to show that

(iii)  Thus the matrix D can be expressed as

where C(t) = [[c_ij(t)]] is a Cauchy matrix for all t ≥ 1. This shows that D is positive definite.

5.6.21  Exercise

The infinite divisibility of the Pascal matrix can be proved in another way as follows. Let λ₁, . . . , λ_n be positive numbers, and let K be the n × n matrix with entries

When λ_j = j, 1 ≤ j ≤ n, this is the Pascal matrix. Use Gauss’s product formula for the gamma function

to see that

Each of the factors in this product is the i, j entry of a matrix that is congruent to a Cauchy matrix. Hence K is infinitely divisible.

Let f be a nonnegative function on . If for each r > 0 the function (f(x))^r is positive definite, then we say that f is an infinitely divisible function. By Schur’s theorem, the product of two infinitely divisible functions is infinitely divisible. If f is a nonnegative function and for each m = 1, 2, . . . the function (f(x))^1/m is positive definite, then f is infinitely divisible.

Some examples of infinitely divisible functions are given in the next few exercises.

5.6.22  Exercise

(i)  The function f(x) = 1/(cosh x) is infinitely divisible.

(ii)  The function f(x) = 1/(cosh x + a) is infinitely divisible for −1 < a ≤ 1. [Hint: Use Exercises 1.6.4 and 1.6.5.]

5.6.23  Exercise

In Section 5.2.10 we saw that the function

is positive definite. Using this information and Schur’s theorem one can prove that f is in fact infinitely divisible. The steps of the argument are outlined.

(i)  Let a and b be any two nonnegative real numbers. Then either is positive definite.

Hence

is positive definite.

(ii)  Use the identity

to obtain

(iii)  For 0 < r < 1 we have the expansion

where the coefficients a_n are nonnegative. Use Part (i) of this exercise and of Exercise 5.6.22 to prove that the series above represents a positive definite function. This establishes the assertion that (cosh αx)/(cosh x) is infinitely divisible for 0 < α < 1.

5.6.24  Exercise

The aim of this exercise is to show that the function

is infinitely divisible. Its positive definiteness has been established in Section 5.2.10.

(i)   Imitate the arguments in Exercise 5.6.23. Use the identity

to show that the function

is infinitely divisible for 0 ≤ b ≤ a. (This restriction is needed to handle the term sinh(a − b)x occurring in the series expansion.) This shows that the function (sinh αx)/(sinh x) is infinitely divisible for 1/2 ≤ α ≤ 1.

(ii)  Let α be any number in (0, 1) and choose a sequence

where α_i/α_i+1 ≥ 1/2. Then

Each factor in this product is infinitely divisible, and hence so is the product.

5.6.25  Exercise

(i)   Use Exercise 5.6.24 to show that the function x/(sinh x) is infinitely divisible. [Hint: Take limit α ↓ 0.]

(ii)  Use this and the result of Exercise 5.6.23 to show that the function

is infinitely divisible.

5.6.26  Exercise

Let λ₁, . . . , λ_n be any real numbers. Use the result of Exercise 5.2.22 to show that the matrix

is infinitely divisible. Thus the function f(x) = e^−|x| is infinitely divisible. Use the integral formula

to obtain another proof of this fact.

5.6.27  Exercise

Using the gamma function, as in Exercise 1.6.4, show that for every r > 0

Thus the functions 1/(1 + ix)^r, 1/(1 − ix)^r, and 1/(1 + x²)^r are positive definite for every r > 0. This shows that 1/(1 + x²) is infinitely divisible.

5.6.28  Exercise

Let a and b be nonnegative numbers with a ≥ b. Let 0 < r < 1. Use the integral formula (1.39) to show that

where µ is a positive measure. This is equal to

Show that this is positive definite as a function of x. Note that it suffices to show that for each λ > 0,

is positive definite. This, in turn, follows from the integral representation

where γ = [(aλ + b)/(1 + λ)]^1/2. Thus, for a ≥ b the function f(x) = (1 + bx²)/(1 + ax²) is infinitely divisible.

5.6.29  Exercise

Show that the function f(x) = (tanh x)/x is infinitely divisible. [Hint: Use the infinite product expansion for f(x).]

5.6.30  Exercise

Let t > −1 and consider the function

Use the identity

to obtain the equality

Use the binomial theorem and Exercise 5.6.29 to prove that f is infinitely divisible for −1 < t ≤ 1.

Thus many of the positive definite functions from Section 5.2 are infinitely divisible. Consequently the associated positive definite matrices are infinitely divisible. In particular, for any positive numbers λ₁, . . . , λ_n the n × n matrices V, W and Y whose entries are, respectively,

are infinitely divisible.

5.6.31  Another proof of Bochner’s Theorem

The reader who has worked her way through the theory of Pick functions (as given in Chapter V of MA) may enjoy the proof outlined below.

(i)  Let φ be a positive definite function on , continuous at 0. Let z = x + iy be a complex number and put

Since φ is bounded, this integral is convergent for y > 0. Thus f is an analytic function on the open upper half-plane H₊.

(ii)  Observe that

and so from (5.60) we have

First substitute u = t + v in both the integrals, and then interchange u and v in the second integral to obtain

Observe that these two double integrals are over the quarter-planes {(u, v) : u ≥ v ≥ 0} and {(u, v) : v ≥ u ≥ 0}, respectively. Hence

Since φ is a positive definite function, this integral is nonnegative. (Write it as a limit of Riemann sums each of which is nonnegative.)

Thus f maps the upper half-plane into the right half-plane. So i f(z) is a Pick function.

(iii)  For η > 0

Hence, by Problem V.5.9 of MA, there exists a finite positive measure µ on such that

(iv)  Thus we have

(v)  Compare the expression for f in (5.60) with the one obtained in (iv) and conclude

This is the assertion of Bochner’s theorem.

5.7  NOTES AND REFERENCES

Positive definite functions have applications in almost every area of modern analysis. In 1907 C. Carathéodory studied functions with power series

and found necessary and sufficient conditions on the sequence {a_n} in order that f maps the unit disk into the right half-plane. In 1911 O. Toeplitz observed that Carathéodory’s condition is equivalent to (5.1). The connection with Fourier series and transforms has been pointed out in this chapter. In probability theory positive definite functions arise as characteristic functions of various distributions. See E. Lukacs, Characteristic Functions, Griffin, 1960, and R. Cuppens, Decomposition of Multivariate Probabilities, Academic Press, 1975. We mention just one more very important area of their application: the theory of group representations.

Let G be a locally compact topological group. A (continuous) complex-valued function φ on G is positive definite if for each N = 1, 2, . . . , the N × N matrix is positive for every choice of elements g₀, . . . , g_N−1 from G. A unitary representation of G is a homomorphism from G into the group of unitary operators on a Hilbert space H such that for every fixed the map from G into is continuous. (This is called strong continuity.) It is easy to see that if U_g is a unitary representation of G in the Hilbert space , then for every the function

is positive definite on G. (This is a generalization of Exercise 5.6.1.) The converse is an important theorem of Gelfand and Raikov proved in 1943. This says that for every positive definite function φ on G there exist a Hilbert space , a unitary representation U_g of G in , and a vector such that the equation (5.61) is valid. This is one of the first theorems in the representation theory of infinite groups. One of its corollaries is that every locally compact group has sufficiently many irreducible unitary representations. More precisely, for every element g of G different from the identity, there exists an irreducible unitary representation of G for which U_g is not the identity operator.

An excellent survey of positive definite functions is given in J. Stewart, Positive definite functions and generalizations, an historical survey, Rocky Mountain J. Math., 6 (1976) 409–434. Among books, we recommend C. Berg, J.P.R. Christensen, and P. Ressel, Harmonic Analysis on Semigroups, Springer, 1984, and Z. Sasvári, Positive Definite and Definitizable Functions Akademie-Verlag, Berlin, 1994.

In Section 5.2 we have constructed a variety of examples using rather elementary arguments. These, in turn, are useful in proving that certain matrices are positive. The criterion in 5.2.15 is due to G. Pólya, Remarks on characteristic functions, Proc. Berkeley Symp. Math. Statist. & Probability, 1949, pp.115-123. This criterion is very useful as its conditions can be easily verified.

The ideas and results of Sections 5.2 and 5.3 are taken from the papers R. Bhatia and K. R. Parthasarathy, Positive definite functions and operator inequalities, Bull. London Math. Soc. 32 (2000) 214–228, H. Kosaki, Arithmetic-geometric mean and related inequalities for operators, J. Funct. Anal., 15 (1998) 429–451, F. Hiai and H. Kosaki, Comparison of various means for operators, ibid., 163 (1999) 300–323, and F. Hiai and H. Kosaki, Means for matrices and comparison of their norms, Indiana Univ. Math. J., 48 (1999) 899–936.

The proof of Theorem 5.3.1 given here is from R. Bhatia and K. B. Sinha, Derivations, derivatives and chain rules, Linear Algebra Appl., 302/303 (1999) 231–244. Theorem 5.3.3 was proved by K. Löwner (C. Loewner) in Über monotone Matrixfunctionen, Math. Z., 38 (1934) 177–216. Loewner then used this theorem to show that a function is operator monotone on the positive half-line if and only if it has an analytic continuation mapping the upper half-plane into itself. Such functions are characterized by certain integral representations, namely, f is operator monotone if and only if

for some real numbers α and β with β ≥ 0, and a positive measure µ that makes the integral above convergent. The connection between positivity of Loewner matrices and complex functions is made via Carathéodory’s theorem (mentioned at the beginning of this section) and its successors. Following Loewner’s work operator monotonicity of particular examples such as 5.3.5–5.3.8 was generally proved by invoking the latter two criteria (analytic continuation or integral representation). The more direct proofs based on the positivity of Loewner matrices given here are from the 2000 paper of Bhatia and Parthasarathy.

The inequality (5.24) and the more general (5.28) were proved in R. Bhatia and C. Davis, More matrix forms of the arithmetic-geometric mean inequality, SIAM J. Matrix Anal. Appl., 14 (1993) 132–136. For the operator norm alone, the inequality (5.28) was proved by E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann., 123 (1951) 415–438. The inequality (5.24) aroused considerable interest and several different proofs of it were given by various authors. Two of them, R. A. Horn, Norm bounds for Hadamard products and the arithmetic-geometric mean inequality for unitarily invariant norms, Linear Algebra Appl., 223/224 (1995) 355–361, and R. Mathias, An arithmetic-geometric mean inequality involving Hadamard products, ibid., 184 (1993) 71–78, observed that the inequality follows from the positivity of the matrix in (5.27). The papers by Bhatia-Parthasarathy and Kosaki cited above were motivated by extending this idea further. The two papers used rather similar arguments and obtained similar results. The program was carried much further in the two papers of Hiai and Kosaki cited above to obtain an impressive variety of results on means. The interested reader should consult these papers as well as the monograph F. Hiai and H. Kosaki, Means of Hilbert Space Operators, Lecture Notes in Mathematics Vol. 1820, Springer, 2003.

The theorems of Herglotz and Bochner concern the groups and . They were generalized to locally compact abelian groups by A. Weil, by D. A. Raikov, and by A. Powzner, in independent papers appearing almost together. Further generalizations (non-abelian or non-locally compact groups) exist. The original proof of Bochner’s theorem appears in S. Bochner, Vorlesungen über Fouriersche Integrale, Akademie-Verlag, Berlin, 1932. Several other proofs have been published. The one given in Section 5.5 is taken from R. Goldberg, Fourier Transforms, Cambridge University Press, 1961, and that in Section 5.6 from N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, 1993 (reprint of original editions). A generalization to distributions is given in L. Schwartz, Théorie des Distributions, Hermann, 1954.

Integral representations such as the one given by Bochner’s theorem are often viewed as a part of “Choquet Theory.” Continuous positive definite functions φ(x) such that φ(0) = 1 form a compact convex set; the family is the set of extreme points of this convex set.

Exercise 5.6.1 is an adumbration of the connections between positive definite functions and spectral theory of operators. A basic theorem of M. H. Stone in the latter subject says that every unitary representation of R in a Hilbert space H is of the form U_t = e^itA for some (possibly unbounded) self-adjoint operator A. (The operator A is bounded if and only if ||U_t − I|| → 0 as t → 0.) The theorems of Stone and Bochner can be derived from each other. See M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. I, II, Academic Press, 1972, 1975, Chapters VIII, IX.

A sequence is of positive type if for every positive integer N, we have

for every finite sequence of complex numbers ξ₀, ξ₁, . . . , ξ_N−1. This is equivalent to the requirement that for each N = 1, 2, . . . , the N × N matrix

is positive. Compare these conditions with (5.1) and (5.2). (Matrices of the form (5.64) are called Hankel matrices while those of the form (5.2) are Toeplitz matrices.) A complex valued function φ on the positive half-line [0, ∞) is of positive type if for each N the N × N matrix

is positive for every choice of x₀, . . . , x_N−1 in [0, ∞). A theorem of Bernstein and Widder says that φ is of positive type if and only if there exists a positive measure µ on [0, ∞) such that

i.e., φ is the Laplace transform of a positive measure µ. Such functions are characterized also by being completely monotone, which, by definition, means that

See MA p.148 for the connection such functions have with operator monotone functions. The book of Berg, Christensen, and Ressel cited above is a good reference for the theory of these functions.

Our purpose behind this discussion is to raise a question. Suppose f is a function mapping [0, ∞) into itself. Say that f is in the class if for each N the matrix

is positive for every choice λ₁, . . . , λ_N in [0, ∞). The class is precisely the operator monotone functions. Is there a good characterisation of functions in ? One can easily see that if , then so does 1/f. It is known that is contained in ; see, e.g., M. K. Kwong, Some results on matrix monotone functions, Linear Algebra Appl., 118 (1989) 129–153. (It is easy to see, using the positivity of the Cauchy matrix, that for every λ > 0 the function g(t) = λt/(λ + t) is in . The integral representation (5.62) then shows that every function in is in .)

The conjecture stated after Exercise 5.6.3 goes back to D. Bessis, P. Moussa, and M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics, J. Math. Phys., 16 (1975) 2318–2325. A more recent report on the known partial results may be found in P. Moussa, On the representation of Tr as a Laplace transform, Rev. Math. Phys., 12 (2000) 621–655. E. H. Lieb and R. Seiringer, Equivalent forms of the Bessis-Moussa-Villani conjecture, J. Stat. Phys., 115 (2004) 185–190, point out that the statement of this conjecture is equivalent to the following: for all A and B positive, and all natural numbers p, the polynomial has only positive coefficients. When this polynomial is multiplied out, the co-efficient of λ^r is a sum of terms each of which is the trace of a word in A and B. It has been shown by C. R. Johnson and C. J. Hillar, Eigenvalues of words in two positive definite letters, SIAM J. Matrix Anal. Appl., 23 (2002) 916-928, that some of the individual terms in this sum can be negative. For example, tr A²B²AB can be negative even when A and B are positive.

The matrix Z in Exercise 5.6.7 was studied by M. K. Kwong, On the definiteness of the solutions of certain matrix equations, Linear Algebra Appl., 108 (1988) 177–197. It was shown here that for each n ≥ 2, there exists a number t_n such that Z is positive for all t in (−2, t_n], and further t_n > 2 for all n, t_n = ∞, 8, 4 for n = 2, 3, 4, respectively. The complete solution (given in Exercise 5.6.7) appears in the 2000 paper of Bhatia-Parthasarathy cited earlier. The idea and the method are carried further in R. Bhatia and D. Drissi, Generalised Lyapunov equations and positive definite functions, SIAM J. Matrix Anal. Appl., 27 (2005) 103-295–114. Using a Fourier transforms argument D. Drissi, Sharp inequalities for some operator means, preprint 2006, has shown that the function f(x) = (x cosh α x)/ sinh x is not positive definite when |α| > 1/2. The result of Exercise 5.6.9 is due to E. Andruchow, G. Corach, and D. Stojanoff, Geometric operator inequalities, Linear Algebra Appl., 258 (1997) 295-310, where other related inequalities are also discussed. The result of Exercise 5.6.10 was proved by D. K. Jocić, Cauchy-Schwarz and means inequalities for elementary operators into norm ideals, Proc. Am. Math. Soc., 126 (1998) 2705–2711. Cognate results are proved in D. K. Jocić, Cauchy-Schwarz norm inequalities for weak^∗-integrals of operator valued functions, J. Funct. Anal., 218 (2005) 318–346.

Proposition 5.6.11 is proved in the Hiai-Kosaki papers cited earlier. They also give an example of two means where M₁ ≤ M₂, but M₁ << M₂ is not true. The simple example in Exercise 5.6.12 is from R. Bhatia, Interpolating the arithmetic-geometric mean inequality and its operator version, Linear Algebra Appl., 413 (2006) 355–363.

Conditionally positive definite matrices are discussed in Chapter 4 of the book R. B. Bapat and T. E. S. Raghavan, Nonnegative Matrices and Applications, Cambridge University Press, 1997, and more briefly in Section 6.3 of R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991. This section also contains a succinct discussion of infinitely divisible matrices and references to original papers. The results of Exercises 5.6.20 and 5.6.21 are taken from R. Bhatia, Infinitely divisible matrices, Am. Math. Monthly, 113 (2006) 221–235, and those of Exercises 5.6.23, 5.6.24 and 5.6.25 from R. Bhatia and H. Kosaki, Mean matrices and infinite divisibility, preprint 2006. In this paper it is shown that for several classes of means m(a, b), matrices of the form [[m(λ_i, λ_j)]] are infinitely divisible if for all a and b; and if , then matrices of the form [[1/m(λ_i, λ_j)]] are infinitely divisible. The contents of Exercise 5.6.28 are taken from H. Kosaki, On infinite divisibility of positive definite functions, preprint 2006. In this paper Kosaki uses very interesting ideas from complex analysis to obtain criteria for infinite divisibility.

We have spoken of Loewner’s theorems that say that the Loewner matrices associated with a function f on [0, ∞) are positive if and only if f has an analytic continuation mapping the upper half-plane into itself. R. A. Horn, On boundary values of a schlicht mapping, Proc. Am. Math. Soc., 18 (1967) 782–787, showed that this analytic continuation is a one-to-one (schlicht) mapping if and only if the Loewner matrices associated with f are infinitely divisible. This criterion gives another proof of the infinite divisibility of the matrices in Sections 5.3.5 and 5.3.7.

Infinitely divisible distribution functions play an important role in probability theory. These are exactly the limit distributions for sums of independent random variables. See, for example, L. Breiman, Probability, Addison-Wesley, 1968, pp. 190–196, or M. Loeve, Probability Theory, Van Nostrand, 1963, Section 22. The two foundational texts on this subject are B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, 1954, and P. Lévy, Théorie de l’ Addition des Variables Aléatoires, Gauthier-Villars, 1937. The famous LévyKhintchine Formula says that a continuous positive definite function φ, with φ(0) = 1, is infinitely divisible if and only if it can be represented as

where γ is a real number, G is a nondecreasing function of bounded variation, and the integrand at u = 0 is to be interpreted as the limiting value −t²/2. This representation of log f(t) is unique. See the book by Gnedenko and Kolmogorov, p. 76. A list of such representations for some of the examples we have considered is given in Lukacs, p. 93.

A positive definite function is said to be self-decomposable if for every real number ν in (0, 1) there exists a positive definite function φ_ν such that

Such functions are of importance in probability theory; see Section 23.3 of Loeve’s book. It is shown there that if φ is self-decomposable, then it is infinitely divisible, and so are the functions φ_ν, 0 < ν < 1. Our discussion in Section 5.2.10 shows that the functions 1/(1 + x²), 1/cosh x and x/sinh x are self-decomposable. So, the infinite divisibility of some of the functions in Exercises 5.6.22–5.6.28 is a consequence also of Loeve’s theorem.

In Exercise 5.6.28 we observed that if a ≥ b ≥ 0, then the function f(x) = (1 + bx²)/(1 + ax²) is infinitely divisible, and in Exercise 5.6.29 asked the reader to use this to show that g(x) = (tanh x)/x is infinitely divisible. In the paper Z. J. Jurek and M. Yor, Self decomposable laws associated with hyperbolic functions, Probability and Mathematical Statistics, 24 (2004), 181–190, it is observed that the function f is not self-decomposable but the function g is. Some of the results in H. Kosaki, On infinite divisibility of positive definite functions, preprint 2006, can be rephrased to say that certain functions are self-decomposable. His list includes, for example, 1/(cosh x + a), −1 ≤ a ≤ 1. The relevance and importance of these functions in probability theory is explained in the paper of Jurek and Yor, and in the references therein.

Finally, we make a few comments on operator inequalities with special reference to the arithmetic-geometric mean inequality. Operator inequalities are sought and found in three different versions. If A and B are positive, we may have sometimes an inequality A ≥ B. By Weyl’s monotonicity principle, this implies that λ_j(A) ≥ λ_j(B), 1 ≤ j ≤ n, where λ_j(A) is the jth eigenvalue of A counted in decreasing order. (This is equivalent to the existence of a unitary matrix U such that A ≥ UBU^∗.) This, in turn implies that |||A||| ≥ |||B||| for all unitarily invariant norms. If A and B are not positive, we may ask whether |A| ≥ |B|. This implies the set of inequalities s_j(A) ≥ s_j(B) for all the singular values, which in turn implies |||A||| ≥ |||B||| for all unitarily invariant norms.

In Chapter 4 we saw an arithmetic-geometric mean inequality of the first kind; viz., for any two positive matrices. The inequality (5.23) and its stronger version (5.24) are of the third kind. An inequality of the second kind was proved by R. Bhatia and F. Kittaneh, On the singular values of a product of operators, SIAM J. Matrix Anal. Appl., 11 (1990) 272–277. This says that , for 1 ≤ j ≤ n. This inequality for singular values implies the inequality (5.23) for norms. A stronger version s_j(A^1/2XB^1/2) ≤ s_j(AX + XB)/2 is not always true. So, there is no second level inequality generalising (5.24).

For positive numbers a and b, the arithmetic-geometric mean inequality may be written in three different ways:

(i)  ,

(ii)  ,

(iii)  .

While each of these three may be obtained from the other, the matrix versions suggested by them are different. For example (i) leads to the question whether

This is different from the inequality of Bhatia and Kittaneh stated above. It is not known whether this is true when n > 2. Weaker than this is the third-level inequality

This too is known to be true for a large class of unitarily invariant norms (including Schatten p-norms for p = 1 and for p ≥ 2). It is not known whether it is true for all unitarily invariant norms. From properties of the matrix square function, one can see that this last (unproven) inequality is stronger than the assertion

This version of the arithmetic-geometric mean inequality is known to be true. Thus there are quite a few subtleties involved in noncommutative versions of simple inequalities. A discussion of some of these matters may be found in R. Bhatia and F. Kittaneh, Notes on matrix arithmetic-geometric mean inequalities, Linear Algebra Appl., 308 (2000) 203–211, where the results just mentioned are proved. We recommend the monograph X. Zhan, Matrix Inequalities, Lecture Notes in Mathematics Vol. 1790, Springer, 2002 for a discussion of several topics related to these themes.