Chapter Two

---

---

Positive Linear Maps

In this chapter we study linear maps on spaces of matrices. We use the symbol Φ for a linear map from into . When k = 1 such a map is called a linear functional, and we use the lower-case symbol φ for it. The norm of Φ is

In general, it is not easy to calculate this. One of the principal results of this chapter is that if Φ carries positive elements of to positive elements of , then ||Φ|| = ||Φ(I)||.

2.1  REPRESENTATIONS

The interplay between algebraic properties of linear maps Φ and their metric properties is best illustrated by considering representations of in . These are linear maps that

(i)   preserve products; i.e., Φ(AB) = Φ(A)Φ(B);

(ii)   preserve adjoints; i.e., Φ(A^∗) = Φ(A)^∗;

(iii)  preserve the identity; i.e., Φ(I) = I.

Let σ(A) denote the spectrum of A, and spr(A) its spectral radius.

2.1.1  Exercise

If Φ has properties (i) and (iii), then

Hence

Our norm || · || has two special properties related to the ∗ operation: ||A||² = ||A^∗A||; and ||A|| = spr(A) if A is Hermitian. So, if Φ is a representation we have

Thus ||Φ(A)|| ≤ ||A|| for all A. Since Φ(I) = I, we have ||Φ|| = 1.

We have shown that every representation has norm one.

How does one get representations? For each unitary element of M_n, Φ(A) = U^∗AU is a representation. Direct sums of such maps are representations; i.e., if U₁, . . . , U_r are n × n unitary matrices, then is a representation.

Choosing U_j = I, 1 ≤ j ≤ r, we get the representation Φ(A) = . The operator is unitarily equivalent to , and is another representation.

2.1.2  Exercise

All representations of are obtained by composing unitary conjugations and tensor products with I_r, r = 1, 2, . . .. Thus we have exhausted the family of representations by the examples we saw above. [Hint: A representation carries orthogonal projections to orthogonal projections, unitaries to unitaries, and preserves unitary conjugation.]

Thus the fact that ||Φ|| = 1 for every representation Φ is not too impressive; we do know and ||U^∗AU|| = ||A||.

We will see how we can replace the multiplicativity condition (i) by less restrictive conditions and get a richer theory.

2.2  POSITIVE MAPS

A linear map is called positive if Φ(A) ≥ O whenever A ≥ O. It is said to be unital if Φ(I) = I. We will say Φ is strictly positive if Φ(A) > O whenever A > O. It is easy to see that a positive linear map Φ is strictly positive if and only if Φ(I) > O.

2.2.1  Examples

(i)   φ(A) = trA is a positive linear functional; is positive and unital.

(ii)   Every linear functional on M_n has the form φ(A) = trAX for some . It is easy to see that φ is positive if and only if X is a positive matrix; φ is unital if trX = 1. (Positive matrices of trace one are called density matrices in the physics literature.)

(iii)  Let , the sum of all entries of A. If e is the vector with all of its entries equal to one, and E = ee^∗, the matrix with all entries equal to one, then

Thus φ is a positive linear functional.

(iv)  The map is a positive map of into itself. (Its range consists of scalar matrices.)

(v)   Let A^tr denote the transpose of A. Then the map Φ(A) = A^tr is positive.

(vi)  Let X be an n × k matrix. Then Φ(A) = X^∗AX is a positive map from into .

(vii)  A special case of this is the compression map that takes an n× n matrix to a k × k block in its top left corner.

(viii)  Let P₁, . . . , P_r be mutually orthogonal projections with P_r = I. The operator is called a pinching of A. In an appropriate coordinate system this can be described as

Every pinching is positive. A special case of this is r = n and each P_j is the projection onto the linear span of the basis vector e_j. Then C(A) is the diagonal part of A.

(ix)  Let B be any positive matrix. Then the map is positive. So is the map Φ(A) = A ◦ B.

(x)  Let A be a matrix whose spectrum is contained in the open right half plane. Let L_A(X) = A^∗X + XA. The operator L_A on is invertible and its inverse is a positive linear map. (See the discussion in Exercise 1.2.10.)

(xi)  Any positive linear combination of positive maps is positive. Any convex combination of positive unital maps is positive and unital.

It is instructive to think of positive maps as noncommutative (matrix) averaging operations. Let C(X) be the space of continuous functions on a compact metric space. Let φ be a linear functional on C(X). By the Riesz representation theorem, there exists a signed measure µ on X such that

The linear functional φ is called positive if φ(f) ≥ 0 for every (pointwise) nonnegative function f. For such a φ, the measure µ representing it is a positive measure. If φ maps the function f ≡ 1 to the number 1, then φ is said to be unital, and then µ is a probability measure. The integral (2.3) is then written as

and called the expectation of f. Every positive, unital, linear functional on C(X) is an expectation (with respect to a probability measure µ). A positive, unital, linear map Φ may thus be thought of as a noncommutative analogue of an expectation map.

2.3  SOME BASIC PROPERTIES OF POSITIVE MAPS

We prove three theorems due to Kadison, Choi, and Russo and Dye. Our proofs use 2 × 2 block matrix arguments.

2.3.1  Lemma

Every positive linear map is adjoint-preserving; i.e., Φ(T^∗) = Φ(T)^∗ for all T.

Proof.   First we show that Φ(A) is Hermitian if A is Hermitian. Every Hermitian matrix A has a Jordan decomposition

So,

is the difference of two positive matrices, and is therefore Hermitian. Every matrix T has a Cartesian decomposition

So,

2.3.2  Theorem ( Kadison’s Inequality)

Let Φ be positive and unital. Then for every Hermitian A

Proof.   By the spectral theorem, , where λ_j are the eigenvalues of A and P_j the corresponding projections with . Then and

Since P_j are positive, so are Φ(P_j). Therefore,

Each summand in the last sum is positive and, hence, so is the sum. By Theorem 1.3.3, therefore,

2.3.3  Exercise

The inequality (2.5) may not be true if Φ is not unital.

Recall that for real functions we have (Ef)² ≤ Ef². The inequality (2.5) is a noncommutative version of this. It should be pointed out that not all inequalities for expectations of real functions have such noncommutative counterparts. For example, we do have (Ef)⁴ ≤ Ef⁴, but the analogous inequality Φ(A)⁴ ≤ Φ(A⁴) is not always true. To see this, let Φ be the compression map from to , taking a 3 × 3 matrix to its top left 2 × 2 submatrix. Let

Then and .

This difference can be attributed to the fact that while the function f(t) = t⁴ is convex on the real line, the matrix function f(A) = A⁴ is not convex on Hermitian matrices.

The following theorem due to Choi generalizes Kadison’s inequality to normal operators.

2.3.4  Theorem (Choi)

Let Φ be positive and unital. Then for every normal matrix A

Proof.   The proof is similar to the one for Theorem 2.3.2. We have

So

is positive.     ■

In Chapter 3, we will see that the condition that A be normal can be dropped if we impose a stronger condition (2-positivity) on Φ.

2.3.5  Exercise

If A is normal, then Φ(A) need not be normal. Thus the left-hand sides of the two inequalities (2.6) can be different.

2.3.6  Theorem (Choi’s Inequality)

Let Φ be strictly positive and unital. Then for every strictly positive matrix A

Proof.   The proof is again similar to that of Theorem 2.3.2. Now we have with λ_j > 0. Then , and

is positive. Hence, by Theorem 1.3.3

2.3.7  Theorem (The Russo-Dye Theorem)

If Φ is positive and unital, then ||Φ|| = 1.

Proof.   We show first that ||Φ(U)|| ≤ 1 when U is unitary. In this case the eigenvalues λ_j are complex numbers of modulus one. So, from the spectral resolution , we get

Hence, by Proposition 1.3.1, ||Φ(U)|| ≤ 1. Now if A is any contraction, then we can write where U, V are unitary. (Use the singular value decomposition of A and observe that if 0 ≤ s ≤ 1, then we have for some θ.) So

Thus ||Φ|| ≤ 1, and since Φ is unital ||Φ|| = 1.     ■

Second proof.  Let ||A|| ≤ 1. Then A has a unitary dilation Â

(Check that this is a unitary element of M_2n.)

Now let Ψ be the compression map taking a 2n × 2n matrix to its top left n × n corner. Then Ψ is positive and unital. So, the composition Φ ◦ Ψ is positive and unital. Now Choi’s inequality (2.6) can be used to get

But this says

This shows that ||Φ(A)|| ≤ 1 whenever ||A|| ≤ 1. Hence, ||Φ|| = 1.     ■

We can extend the result to any positive linear map as follows.

2.3.8  Corollary

Let Φ be a positive linear map. Then ||Φ|| = ||Φ(I)||.

Proof.   Let P = Φ(I), and assume first that P is invertible. Let

Then Ψ is a positive unital linear map. So, we have

Thus ||Φ|| ≤ ||P||; and since Φ(I) = P, we have ||Φ|| = ||P||. This proves the assertion when Φ(I) is invertible. The general case follows from this by considering the family Φ_ε(A) = Φ(A) + εI and letting ε ↓ 0.     ■

The assertion of (this Corollary to) the Russo-Dye theorem is some times phrased as: every positive linear map on attains its norm at the identity matrix.

2.3.9  Exercise

There is a simpler proof of this theorem in the case of positive linear functionals. In this case φ(A) = trAX for some positive matrix X. Then

Here ||T||₁ is the trace norm of T defined as ||T||₁ = s₁(T)+· · ·+s_n(T). The inequality above is a consequence of the fact that this norm is the dual of the norm || · ||.

2.4  SOME APPLICATIONS

We have seen several examples of positive maps. Using the Russo-Dye Theorem we can calculate their norms easily. Thus, for example,

for every pinching of A. (This can be proved in several ways. See MA pp. 50, 97.)

If A is positive, then the Schur multiplier S_A is a positive map. So,

This too can be proved in many ways. We have seen this before in Theorem 1.4.1.

We have discussed the Lyapunov equation

where A is an operator whose spectrum is contained in the open right half plane. (Exercise 1.2.10, Example 2.2.1 (x)). Solving this equation means finding the inverse of the Lyapunov operator L_A defined as L_A(X) = A^∗X + XA. We have seen that is a positive linear map. In some situations W is known with some imprecision, and we have the perturbed equation

If X and X+ΔX are the solutions to (2.11) and (2.12), respectively, one wants to find bounds for ||ΔX||. This is a very typical problem in numerical analysis. Clearly,

Since is positive we have . This simplifies the problem considerably. The same considerations apply to the Stein equation (Exercise 1.2.11).

Let be the k-fold tensor product and let be the k-fold product of an operator A on . For 1 ≤ k ≤ n, let be the subspace of spanned by antisymmetric tensors. This is called the antisymmetric tensor product, exterior product, or Grassmann product. The operator leaves this space invariant and the restriction of to it is denoted as ∧^kA. This is called the kth Grassmann power, or the exterior power of A.

Consider the map . The derivative of this map at A, denoted as , is a linear map from into . Its action is given as

Hence,

It follows that

We want to find an expression for ||D ∧^k (A)||.

Recall that ∧^k is multiplicative, ∗ - preserving, and unital (but not linear!). Let A = USV be the singular value decomposition of A. Then

Thus

and hence

Thus to calculate ||D ∧^k (A)||, we may assume that A is positive and diagonal.

Now note that if A is positive, then for every positive B, the expression (2.13) is positive. So is a positive linear map from into . The operator D ∧^k (A)(B) is the restriction of (2.13) to the invariant subspace . So ∧^k(A) is also a positive linear map. Hence

Let A = diag(s₁, . . . , s_n) with s₁ ≥ s₂ ≥ · · · ≥ s_n ≥ 0. Then ∧^kA is a diagonal matrix of size whose diagonal entries are s_i1s_i2 · · · s_ik, 1 ≤ i₁ < i₂ < · · · < i_k ≤ n. Use this to see that

the elementary symmetric polynomial of degree k − 1 with arguments s₁, . . . , s_k.

The effect of replacing D ∧^k (A)(B) by D ∧^k (A)(I) is to reduce a highly noncommutative problem to a simple commutative one. Another example of this situation is given in Section 2.7.

2.5  THREE QUESTIONS

Let be a linear map. We have seen that if Φ is positive, then

Clearly, this is a useful and important theorem. It is natural to explore how much, and in what directions, it can be extended.

Question 1   Are there linear maps other than positive ones for which (2.16) is true? In other words, if a linear map Φ attains its norm at the identity, then must Φ be positive?

Before attempting an answer, we should get a small irritant out of the way. If the condition (2.16) is met by Φ, then it is met by −Φ also. Clearly, both of them cannot be positive maps. So assume Φ satisfies (2.16) and

2.5.1  Exercise

If k = 1, the answer to our question is yes. In this case φ(A) = trAX for some X. Then ||φ|| = ||X||₁ (see Exercise 2.3.9). So, if φ satisfies (2.16) and (2.17), then ||X||₁ = trX. Show that this is true if and only if X is positive. Hence φ is positive.

If k ≥ 2, this is no longer true. For example, let Φ be the linear map on defined as

Then ||Φ|| = ||Φ(I)|| = 1 and Φ(I) ≥ O, but Φ is not positive.

It is a remarkable fact that if Φ is unital and ||Φ|| = 1, then Φ is positive. Thus supplementing (2.16) with the condition Φ(I) = I ensures that Φ is positive. This is proved in the next section.

Question 2   Let S be a linear subspace of and let be a linear map. Do we still have a theorem like the Russo-Dye theorem? In other words how crucial is the fact that the domain of Φ is M_n (or possibly a subalgebra)?

Again, for the question to be meaningful, we have to demand of S a little more structure. If we want to talk of positive unital maps, then S must contain some positive elements and I.

2.5.2  Definition

A linear subspace S of is called an operator system if it is ∗ closed (i.e., if A ∈ S, then A^∗ ∈ S ) and contains I.

Let S be an operator system. We want to know whether a positive linear map attains its norm at I. The answer is yes if k = 1, and no if k ≥ 2. However, we do have ||Φ|| ≤ 2||Φ(I)|| for all k.

A related question is the following:

Question 3   By the Hahn-Banach theorem, every linear functional φ on (a linear subspace) S can be extended to a linear functional on in such a way that . Now we are considering positivity rather than norms. So we may ask whether a positive linear functional φ on an operator system S in can be extended to a positive linear functional on . The answer is yes. This is called the Krein extension theorem. Then since , we have ||φ|| = φ(I).

Next we may ask whether a positive linear map Φ from into can be extended to a positive linear map from into . If this were the case, then we would have ||Φ|| = ||Φ(I)||. But we have said that this is not always true when k ≥ 2. This is illustrated by the following example.

2.5.3  Example

Let n be any number bigger than 2 and let S be the n×n permutation matrix

Let S be the collection of all matrices C of the form C = aI+bS+cS^∗, a, b, c ∈ . (The matrices C are circulant matrices.) Then is an operator system in . What are the positive elements of ? First, we must have a ≥ 0 and . The eigenvalues of S are 1, ω, . . . , ωⁿ⁻¹, the n roots of 1. So, the eigenvalues of C are

and C is positive if and only if all these numbers are nonnegative.

It is helpful to consider the special case n = 4. The fourth roots of unity are {1, i, −1, −i} and one can see that a matrix C of the type above is positive if and only if

Let be the map defined as

Then Φ is linear, positive, and unital. Since

. So, Φ cannot be extended to a positive, linear, unital map from into .

2.5.4  Exercise

Let n ≥ 3 and consider the operator system defined in the example above. For every t the map defined as

is linear and unital. Show that for 1 < t < 2 there exists an n such that the map Φ is positive.

We should remark here that the elements of commute with each other (though, of course, is not a subalgebra of ).

In the next section we prove the statements that we have made in answer to the three questions.

2.6  POSITIVE MAPS ON OPERATOR SYSTEMS

Let be an operator system in _. the set of self-adjoint elements of , and the set of positive elements in it.

Some of the operations that we performed freely in may take us outside . Thus if , then Re and Im T = are in . However, if , then the positive and negative parts A_± in the Jordan decomposition of A need not be in . For example, consider

This is an operator system. The matrix is in . Its Jordan components are

They do not belong to S.

However, it is possible still to write every Hermitian element A of as

Just choose

Thus we can write every as

Using this decomposition we can prove the following lemma.

2.6.1  Lemma

Let Φ be a positive linear map from an operator system into . Then Φ(T^∗) = Φ(T)^∗ for all .

2.6.2  Exercise

If A = P₁ − P₂ where P₁, P₂ are positive, then

2.6.3  Theorem

Let Φ be a positive linear map from an operator system into . Then

and

(Thus if Φ is also unital, then ||Φ|| = 1 on the space , and ||Φ|| ≤ 2 on .)

Proof.   If P is a positive element of , then O ≤ P ≤ ||P||I, and hence O ≤ Φ(P) ≤ ||P||Φ(I).

If A is a Hermitian element of , use the decomposition (2.18), Exercise 2.6.2, and the observation of the preceding sentence to see that

This proves the first inequality of the theorem. The second is obtained from this by using the Cartesian decomposition of T.     ■

Exercise 2.5.4 shows that the factor 2 in the inequality (ii) of Theorem 2.6.3 is unavoidable in general. It can be dropped when k = 1:

2.6.4  Theorem

Let φ be a positive linear functional on an operator system . Then ||φ|| = φ(I).

Proof.   Let and ||T|| ≤ 1. We want to show |φ(T)| ≤ φ(I). If φ(T) is the complex number re^iθ, we may multiply T by e^−iθ, and thus assume φ(T) is real and positive. So, if T = A+ iB in the Cartesian decomposition, then φ(T) = φ(A). Hence by part (i) of Theorem 2.6.3 φ(T) ≤ φ(I)||A|| ≤ φ(I)||T||.     ■

The converse is also true.

2.6.5  Theorem

Let φ be a linear functional on such that ||φ|| = φ(I). Then φ is positive.

Proof.   Assume, without loss of generality, that φ(I) = 1. Let A be a positive element of and let σ(A) be its spectrum. Let a = min σ(A) and b = max σ(A). We will show that the point φ(A) lies in the interval [a, b]. If this is not the case, then there exists a disk D = {z : |z − z₀| ≤ r} such that φ(A) is outside D but D contains [a, b], and hence also σ(A). From the latter condition it follows that σ(A−z₀I) is contained in the disk {z : |z| ≤ r} , and hence ||A−z₀I|| ≤ r. Using the conditions ||φ|| = φ(I) = 1, we get from this

But then φ(A) could not have been outside D.

This shows that φ(A) is a nonnegative number, and the theorem is proved.     ■

2.6.6  Theorem (The Krein Extension Theorem)

Let be an operator system in . Then every positive linear functional on can be extended to a positive linear functional on .

Proof.   Let φ be a positive linear functional on . By Theorem 2.6.4, ||φ|| = φ(I). By the Hahn-Banach Theorem, φ can be extended to a linear functional on with . But then is positive by Theorem 2.6.5 (or by Exercise 2.5.1).     ■

Finally we have the following theorem that proves the assertion made at the end of the discussion of Question 1 in Section 2.5.

2.6.7  Theorem

Let be an operator system and let _k be a unital linear map such that ||Φ|| = 1. Then Φ is positive.

Proof.   For each unit vector x in , let

This is a unital linear functional on . Since |φ_x(A)| ≤ ||Φ(A)|| ≤ ||A||, we have ||φ_x|| = 1. So, by Theorem 2.6.5, φ_x is positive. In other words, if A is positive, then for every unit vector x

But that means Φ is positive.     ■

2.7  SUPPLEMENTARY RESULTS AND EXERCISES

Some of the theorems in Section 2.3 are extended in various directions in the following propositions.

2.7.1  Proposition

Let Φ be a positive unital linear map on and let A be a positive matrix. Then

Proof.   Let 0 < r < 1. Using the integral representation (1.39) we have

where µ is a positive measure on (0, ∞). So it suffices to show that

for all λ > 0. We have the identity

Apply Φ to both sides and use Theorem 2.3.6 to get

The identity stated above shows that the last expression is equal to Φ(A)(λ + Φ(A))⁻¹.     ■

2.7.2  Exercise

Let Φ be a positive unital linear map on and let A be a positive matrix. Show that

if 1 ≤ r ≤ 2. If A is strictly positive, then this is true also when −1 ≤ r ≤ 0. [Hint: Use integral representations of A^r as in Theorem 1.5.8, Exercise 1.5.10, and the inequalities (2.5) and (2.7).]

2.7.3  Proposition

Let Φ be a strictly positive linear map on M_n. Then

whenever H is Hermitian and A > 0.

Proof.   Let

Then Ψ is positive and unital. By Kadison’s inequality we have Ψ(Y ²) ≥ Ψ(Y )² for every Hermitian Y . Choose Y = A^−1/2HA^−1/2 to get

Use (2.21) now to get (2.20).     ■

2.7.4  Exercise

Construct an example to show that a more general version of (2.20)

where X is arbitrary and A positive, is not always true.

2.7.5  Proposition

Let Φ be a strictly positive linear map on and let A > O. Then

Proof.   Let Ψ be the linear map defined by (2.21). By the Russo-Dye theorem

Let A ≥ X^∗A⁻¹X and put Y = A^−1/2XA^−1/2. Then Y ^∗Y = A^−1/2 X^∗A⁻¹ XA^−1/2 ≤ I. Hence Ψ(A^−1/2X^∗A^−1/2)Ψ(A^−1/2XA^−1/2) ≤ I. Use (2.21) again to get (2.22).     ■

In classical probability the quantity

is called the variance of the real function f. In analogy we consider

where A is Hermitian and Φ a positive unital linear map on . Kadison’s inequality says var(A) ≥ O. The following proposition gives an upper bound for var(A).

2.7.6  Proposition

Let Φ be a positive unital linear map and let A be a Hermitian operator with mI ≤ A ≤ MI. Then

Proof.   The matrices MI − A and A− mI are positive and commute with each other. So, (MI − A)(A − mI) ≥ O; i.e.,

Apply Φ to both sides and then subtract Φ(A)² from both sides. This gives the first inequality in (2.25). To prove the second inequality note that if m ≤ x ≤ M, then .     ■

2.7.7  Exercise

Let . We say x ≥ 0 if all its coordinates x_j are nonnegative. Let e = (1, . . . , 1).

A matrix S is called stochastic if s_ij ≥ 0 for all i, j, and for all i. Show that S is stochastic if and only if

and

The property (2.26) can be described by saying that the linear map defined by S on is positive, and (2.27) by saying that S is unital.

If x is a real vector, let . Show that if S is a stochastic matrix and m ≤ x_j ≤ M, then

A special case of this is obtained by choosing for all i, j. If , this gives

An inequality complementary to (2.7) is given by the following proposition.

2.7.8  Proposition

Let Φ be strictly positive and unital. Let 0 < m < M. Then for every strictly positive matrix A with mI ≤ A ≤ MI, we have

Proof.   The matrices A − mI and MA⁻¹ − I are positive and commute with each other. So, O ≤ (A − mI)(MA⁻¹ − I). This gives

and hence

Now, if c and x are real numbers, then (c − 2x)² ≥ 0 and therefore, for positive x we have . So, we get

A very special corollary of this is the inequality

for every unit vector x. This is called the Kantorovich inequality.

2.7.9  Exercise

Let f be a convex function on an interval [m, M] and let L be the linear interpolant

Show that if Φ is a unital positive linear map, then for every Hermitian matrix A whose spectrum is contained in [m, M], we have

Use this to obtain Propositions 2.7.6 and 2.7.8.

The space has a natural inner product defined as

If Φ is a linear map on , we define its adjoint Φ^∗ as the linear map that satisfies the condition

2.7.10  Exercise

The linear map Φ is positive if and only if Φ^∗ is positive. Φ is unital if and only if Φ^∗ is trace preserving; i.e., tr Φ^∗(A) = tr A for all A.

We say Φ is a doubly stochastic map on if it is positive,unital, and trace preserving (i.e., both Φ and Φ^∗ are positive and unital).

2.7.11  Exercise

(i)  Let Φ be the linear map on M_n defined as Φ(A) = X^∗AX. Show that Φ^∗(A) = XAX^∗.

(ii)  For any A, let S_A(X) = A◦ X be the Schur product map. Show that (S_A)^∗ = S_A∗.

(iii)  Every pinching is a doubly stochastic map.

(iv)  Let L_A(X) = A^∗X + XA be the Lyapunov operator, where A is a matrix with its spectrum in the open right half plane. Show that (L⁻ _A ¹)^∗ = (L_A∗ )⁻¹.

A norm ||| · ||| on M_n is said to be unitarily invariant if |||UAV ||| = |||A||| for all A and unitary U, V . It is convenient to make a normalisation so that |||A||| = 1 whenever A is a rank-one orthogonal projection.

Special examples of such norms are the Ky Fan norms

and the Schatten p-norms

Note that the operator norm, in this notation, is

and the trace norm is the norm

The norm ||A||₂ is also called the Hilbert-Schmidt norm.

The following facts are well known:

If ||A||_(k) ≤ ||B||_(k) for 1 ≤ k ≤ n, then |||A||| ≤ |||B||| for all unitarily invariant norms. This is called the Fan dominance theorem. (See MA, p. 93.)

For any three matrices A, B, C we have

If Φ is a linear map on and ||| · ||| any unitarily invariant norm, then we use the notation |||Φ||| for

In the same way,

etc.

The norm ||A||₁ is the dual of the norm ||A|| on M_n. Hence

2.7.12  Exercise

Let ||| · ||| be any unitarily invariant norm on M_n.

(i)   Use the relations (2.34) and the Fan dominance theorem to show that if ||Φ|| ≤ 1 and ||Φ^∗|| ≤ 1, then |||Φ||| ≤ 1.

(ii)  If Φ is a doubly stochastic map, then |||Φ||| ≤ 1.

(iii)  If A ≥ O, then |||A ◦ X||| ≤ max a_ii|||X||| for all X.

(iv)  Let L_A be the Lyapunov operator associated with a positively stable matrix A. We know that . Show that in the special case when A is normal we have = [2 min Re λ_i]⁻¹, where λ_i are the eigenvalues of A.

2.7.13  Exercise

Let A and B be Hermitian matrices. Suppose A = Φ(B) for some doubly stochastic map Φ on M_n. Show that A is a convex combination of unitary conjugates of B; i.e., there exist unitary matrices U₁, . . . , U_k and positive numbers p₁, . . . , p_k with such that

[Hints: There exist diagonal matrices D₁ and D₂, and unitary matrices W and V such that A = W^∗D₁W and B = V D₂V ^∗. Use this to show that D₁ = Ψ(D₂) where Ψ is a doubly stochastic map. By Birkhoff’s theorem there exist permutation matrices S₁, . . . , S_k and positive numbers p₁, . . . , p_k with such that

Choose U_j = V S_jW. (Note that the matrices U_j and the numbers p_j depend on Φ, A and B.)]

Let be the set of all n × n Hermitian matrices. This is a real vector space. Let I be an open interval and let C¹(I) be the space of continuously differentiable real functions on I. Let be the set of all Hermitian matrices whose eigenvalues belong to I. This is an open subset of . Every function f in C¹(I) induces a map from into . This induced map is differentiable and its derivative is given by an interesting formula known as the Daleckii-Krein formula.

For each the derivative Df(A) at A is a linear map from into itself. If is the spectral decomposition of A, then the formula is

for every . For i = j, the quotient in (2.38) is to be interpreted as f^′(λ_i).

This formula can be expressed in another way. Let f^[1] be the function on I × I defined as

This is called the first divided difference of f. For , let f^[1](A) be the n × n matrix

where λ₁, . . . , λ_n are the eigenvalues of A. The formula (2.38) says

where ◦ denotes the Schur product taken in a basis in which A is diagonal. A proof of this is given in Section 5.3.

Suppose a real function f on an interval I has the following property: if A and B are two elements of and A ≥ B, then f(A) ≥ f(B). We say that such a function f is matrix monotone of order n on I. If f is matrix monotone of order n for all n = 1, 2, . . . , then we say f is operator monotone.

Matrix convexity of order n and operator convexity can be defined in a similar way. In Chapter 1 we have seen that the function f(t) = t² on the interval [0, ∞) is not matrix monotone of order 2, and the function f(t) = t³ is not matrix convex of order 2. We have seen also that the function f(t) = t^r on the interval [0, ∞) is operator monotone for 0 ≤ r ≤ 1, and it is operator convex for 1 ≤ r ≤ 2 and for −1 ≤ r ≤ 0. More properties of operator monotone and convex functions are studied in Chapters 4 and 5.

It is not difficult to prove the following, using the formula (2.40).

2.7.14  Exercise

If a function f ∈ C¹(I) is matrix monotone of order n, then for each , the matrix f^[1](A) defined in (2.39) is positive.

The converse of this statement is also true. A proof of this is given in Section 5.3. At the moment we note the following interesting consequence of the positivity of f^[1](A).

2.7.15  Exercise

Let f ∈ C¹(I) and let f^′ be the derivative of f. Show that if f is matrix monotone of order n, then for each

By definition

and

This expression is difficult to calculate for functions such as f(t) = t^r, 0 < r < 1. The formula (2.41) gives an easy way to calculate its norm. Its effect is to reduce the supremum in (2.42) to the class of matrices B that commute with A.

2.8  NOTES AND REFERENCES

Since positivity is a useful and interesting property, it is natural to ask what linear transformations preserve it. The variety of interesting examples, and their interpretation as “expectation,” make positive linear maps especially interesting. Their characterization, however, has turned out to be slippery, and for various reasons the special class of completely positive linear maps has gained in importance.

Among the early major works on positive linear maps is the paper by E. Størmer, Positive linear maps of operator algebras, Acta Math., 110 (1963) 233–278. Research expository articles that explain several subtleties include E. Størmer, Positive linear maps of C^∗-algebras, in Foundations of Quantum Mechanics and Ordered Linear Spaces, Lecture Notes in Physics, Vol. 29, Springer, 1974, pp.85–106, and M.-D. Choi, Positive linear maps, in Operator Algebras and Applications, Part 2, R. Kadison ed., American Math. Soc., 1982. Closer to our concerns are Chapter 2 of V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002, and sections of the two reports by T. Ando, Topics on Operator Inequalities, Sapporo, 1978 and Operator-Theoretic Methods for Matrix Inequalities, Sapporo, 1998.

The inequality (2.5) was proved in the paper R. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. Math., 56 (1952) 494–503. This was generalized by C. Davis, A Schwarz inequality for convex operator functions, Proc. Am. Math. Soc., 8 (1957) 42–44, and by M.-D. Choi, A Schwarz inequality for positive linear maps on C^∗-algebras, Illinois J. Math., 18 (1974) 565–574. The generalizations say that if Φ is a positive unital linear map and f is an operator convex function, then we have a Jensen-type inequality

The inequality (2.7) and the result of Exercise 2.7.2 are special cases of this. Using the integral representation of an operator convex function given in Problem V.5.5 of MA, one can prove the general inequality by the same argument as used in Exercise 2.7.2. The inequality (2.43) characterises operator convex functions, as was noted by C. Davis, Notions generalizing convexity for functions defined on spaces of matrices, in Proc. Symposia Pure Math., Vol. VII, Convexity, American Math. Soc., 1963.

In our proof of Theorem 2.3.7 we used the fact that any contraction is an average of two unitaries. The infinite-dimensional analogue says that the unit ball of a C^∗-algebra is the closed convex hull of the unitary elements. (Unitaries, however, do not constitute the full set of extreme points of the unit ball. See P. R. Halmos, A Hilbert Space Problem Book, Second Edition, Springer, 1982.) This theorem about the closed convex hull is also called the Russo-Dye theorem and was proved in B. Russo and H. A. Dye, A note on unitary operators in C^∗-algebras, Duke Math. J., 33 (1966) 413–416.

Applications given in Section 2.4 make effective use of Theorem 2.3.7 in calculating norms of complicated operators. Our discussion of the Lyapunov equation follows the one in R. Bhatia and L. Elsner, Positive linear maps and the Lyapunov equation, Oper. Theory: Adv. Appl., 130 (2001) 107–120. As pointed out in this paper, the use of positivity leads to much more economical proofs than those found earlier by engineers. The equality (2.15) was first proved by R. Bhatia and S. Friedland, Variation of Grassman powers and spectra, Linear Algebra Appl., 40 (1981) 1–18. The alternate proof using positivity is due to V. S. Sunder, A noncommutative analogue of |DX^k| = |kX^k−1|, ibid., 44 (1982) 87-95. The analogue of the formula (2.15) when the antisymmetric tensor product is replaced by the symmetric one was worked out in R. Bhatia, Variation of symmetric tensor powers and permanents, ibid., 62 (1984) 269–276. The harder problem embracing all symmetry classes of tensors was solved in R. Bhatia and J. A. Dias da Silva, Variation of induced linear operators, ibid., 341 (2002) 391–402.

Because of our interest in certain kinds of matrix problems involving calculation or estimation of norms we have based our discussion in Section 2.5 on the relation (2.16). There are far more compelling reasons to introduce operator systems. There is a rapidly developing and increasingly important theory of operator spaces (closed linear subspaces of C^∗-algebras) and operator systems. See the book by V. Paulsen cited earlier, E. G. Effros and Z.-J. Ruan, Operator Spaces, Oxford University Press, 2000, and G. Pisier, Introduction to Operator Space Theory, Cambridge University Press, 2003. This is being called the noncommutative or quantized version of Banach space theory. One of the corollaries of the Hahn-Banach theorem is that every separable Banach space is isometrically isomorphic to a subspace of l_∞; and every Banach space is isometrically isomorphic to a subspace of l_∞(X) for some set X. In the quantized version the commutative space l_∞ is replaced by the noncommutative space where is a Hilbert space. Of course, it is not adequate functional analysis to study just the space l_∞ and its subspaces. Likewise subspaces of are called concrete operator spaces, and then subsumed in a theory of abstract operator spaces.

Our discussion in Section 2.6 borrows much from V. Paulsen’s book. Some of our proofs are simpler because we are in finite dimensions.

Propositions 2.7.3 and 2.7.5 are due to M.-D. Choi, Some assorted inequalities for positive linear maps on C^∗-algebras, J. Operator Theory, 4 (1980) 271–285. Propositions 2.7.6 and 2.7.8 are taken from R. Bhatia and C. Davis, A better bound on the variance, Am. Math. Monthly, 107 (2000) 602–608. Inequalities (2.29), (2.31) and their generalizations are important in statistics, and have been proved by many authors, often without knowledge of previous work. See the article S. W. Drury, S. Liu, C.-Y. Lu, S. Puntanen, and G. P. H. Styan, Some comments on several matrix inequalities with applications to canonical correlations: historical background and recent developments, Sankhyā, Series A, 64 (2002) 453–507.

The Daleckii-Krein formula was presented in Ju. L. Daleckii and S. G. Krein, Formulas of differentiation according to a parameter of functions of Hermitian operators, Dokl. Akad. Nauk SSSR, 76 (1951) 13–16. Infinite dimensional analogues in which the double sum in (2.38) is replaced by a double integral were proved by M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals (English translation), Topics in Mathematical Physics Vol. 1, Consultant Bureau, New York, 1967.

The formula (2.41) was noted in R. Bhatia, First and second order perturbation bounds for the operator absolute value, Linear Algebra Appl., 208/209 (1994) 367–376. It was observed there that this equality of norms holds for several other functions that are not operator monotone. If A is positive and f(A) = A^r, then the equality (2.41) is true for all real numbers r other than those in . This, somewhat mysterious, result was proved in two papers: R. Bhatia and K. B. Sinha, Variation of real powers of positive operators, Indiana Univ. Math. J., 43 (1994) 913–925, and R. Bhatia and J. A. R. Holbrook, Fréchet derivatives of the power function, ibid., 49(2000) 1155–1173. Similar equalities involving higher-order derivatives have been proved in R. Bhatia, D. Singh, and K. B. Sinha, Differentiation of operator functions and perturbation bounds, Commun. Math. Phys., 191 (1998) 603–611.