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5 Connection of the Ideals mathcalT(mathcalH) and mathcalHS(mathcalH) with the Sequence Spaces ell1 and ell2

50

3 Classes of Compact Operators

∞

T =

cn |ψn ϕn |

(3.9)

n=1

where cn ≥ 0, lim cn = 0 and (ϕn ) and (ψn ) are orthonormal sequences in H.

n→∞

(We assume in this and the next section that H is infinite-dimensional; the simpler

finite-dimensional case would only require minor changes in notation.) We denote

as usual

1

= (cn ) ∈ RN

∞

|cn | < ∞ and

2

∞

= (cn ) ∈ RN

|cn |2 < ∞ .

n=1

n=1

The following theorem explains the connections mentioned in the title of this

section.

Theorem 3.10 In the situation of formula (3.9) we have

∞

(a) T ∈ HS(H), if and only if (cn ) ∈

2

, and then T

2

=

cn2

1

2

;

n=1

∞

(b) T ∈ T (H), if and only if (cn ) ∈

1

, and then

T

1

=

cn and tr[T ] =

n=1

∞

cn ϕn |ψn .

n=1

Proof First, suppose that T ∈ HS(H). Choose for H an orthonormal basis K containing the set {ϕn |n ∈ N} (see Theorem 2.21). Then

∞

Tξ

ξ∈K

2

=

∞

cn ψn

2

∞

=

n=1

cn2 , for T ξ =

n=1

cm ϕm |ξ ψm = cn ψn ,

m=1

if ξ = ϕn , and T ξ = 0, if ξ ∈ K \ {ϕn }. This implies (a). Since

1

∞

1

|T | 2 =

cn2 |ϕn ϕn |

n=1

(see Remark 3.1), from the above we see that

the first claim in (b) and the equality T 1 =

∞

ξ∈K ξ| |T |ξ =

n=1 cn , implying

∞

n=1 cn . If T ∈ T (H), we further get

∞

tr T =

ξ|T ξ =

ξ∈K

∞

ϕn |cn ψn =

n=1

cn ϕn |ψn .

n=1

3.5 Connection of the Ideals T (H) and HS (H) with the Sequence Spaces

1

and

2

51

We now list some immediate consequences of the preceding theorem.

Theorem 3.11 (a) If ϕ, ψ ∈ H then |ϕ ψ| ∈ T (H) (⊂ HS(H)) and

|ϕ ψ|

1

=

|ϕ ψ|

2

= ϕ

ψ .

n

n

(b) If T ∈ F(H) then T ∈ T (H). If T =

|ϕi ψi | then tr[T ] =

i=1

ψi |ϕi .

i=1

(c) If P ∈ L(H) is a projection then P ∈ T (H) if and only if P ∈ F(H), and then

tr[P] = dim P(H).

(d) If T ∈ T (H), then the series ∞

n=1 cn |ψn ϕn | in (3.9) converges with respect to

the norm · 1 , and if T ∈ HS(H), then it converges with respect to the norm

· 2.

(e) The set F(H) is dense in T (H) with respect to the norm · 1 and in the set

HS(H) with respect to the norm · 2 .

(f) If T ∈ T (H) then T ≤ T 2 ≤ T 1 .

Proof (a) We may assume that ϕ = 0, ψ = 0. As

|ϕ ψ| = ϕ

−1

ϕ

ψ

ϕ

ψ

−1

ψ,

the claim follows from Theorem 3.10.

(b) If ϕi = 0, ψi = 0, then by Theorem 3.10 (b)

tr |ϕi ψi | = ϕi

ψi

ψi

−1

ϕi

ψi

−1

ϕi = ψi |ϕi ,

and the claim follows form the linearity of the trace.

(c) If P ∈ T (H), then P ∈ C(H) (see Lemma 3.2 (b), Theorem 3.7), so that P ∈

F(H) (Theorem 3.3 (f)). The equality tr P = dim(P(H)) follows from (b) and

Theorem 3.3 (a).

(d) The claims follow from Theorem 3.10, since for example

∞

p

cn |ψn ϕn |

T−

n=1

2

=

cn2

1

2

→ 0, when p → ∞.

n= p+1

(e) This is a direct consequence of (d).

(f) The inequality T

∞

then

n=1

(cn c−1 )2 ≤

∞

n=1

2

≤ T

∞

1 follows from Theorem 3.10, for if 0 = c =

cn c−1 = 1, so that

∞

n=1

cn

n=1

(cn c−1 )2

1

2

≤ 1, and so

52

3 Classes of Compact Operators

∞

cn2

1

2

≤ c.

n=1

The inequality T ≤ T

2

was already seen in Theorem 3.7 (a).

3.6 The Dualities C(H)∗ = T (H) and T (H)∗ = L(H)

The trace class of the Hilbert space H has an important role in operator theory. For

example, equipped with the norm introduced in Definition 3.5, as a normed space

T (H) can be identified with the dual of the space C(H) of compact operators, and

the dual of T (H) in turn with L(H). We prove these results in this section.

Theorem 3.12 Define for each T ∈ T (H) the mapping f T : C(H) → C via the formula f T (S) = tr T S , S ∈ C(H). In this way we obtain a linear isometric bijection

T → f T from the space T (H) (equipped with the norm · 1 ) onto the dual of C(H).

Proof By Theorem 3.8 f T is a linear functional defined on C(H). It follows from Theorems 3.9 and 3.8 that f T is continuous and f T ≤ T 1 . Let now f ∈ C(H)∗ . Since

by Theorem 3.7 HS(H) ⊂ C(H) and S 2 ≥ S for all S ∈ HS(H), the restriction f |HS(H) is a continuous linear functional on the Hilbert space HS(H), so that

by the Fréchet–Riesz theorem there is a T ∈ HS(H) satisfying f (S) = T ∗ |S =

∗

ξ∈K T ξ|Sξ =

ξ∈K ξ|T Sξ = tr T S for all S ∈ HS(H) (here T S ∈ T (H)

by Lemma 3.2 (a)). We prove that T ∈ T (H), f = f T and f T ≥ T 1 . Let

T = V |T | be the polar decomposition of T . Then |T | = V ∗ T and hence by Theorem 3.8

tr S|T | = tr SV ∗ T

= |tr T SV ∗ | = | f (SV ∗ )| ≤ f

S

(3.10)

whenever S ∈ F(H) (implying SV ∗ ∈ F(H) ⊂ T (H)). Let now

∞

|T | =

cn |ϕn ϕn |

n=1

be the representation given by (3.1”) (so that cn ≥ 0). If Pk is the projection onto the

subspace spanned by ϕ1 , . . . , ϕk , then by (3.10) and Theorem 3.10 (b)

k

k

cn = tr

n=1

cn |ϕn ϕn | = tr Pk |T | ≤ f ,

n=1

3.6 The Dualities C (H)∗ = T (H) and T (H)∗ = L(H)

53

so that ∞

k=1 ck < ∞, and by Theorem 3.10 (b) |T | ∈ T (H). Hence T ∈ T (H) and

T 1 = tr |T | ≤ f . The functional f and f T , which are continuous functions on

C(H), agree on the dense subspace F(H), and so f = f T . The mapping T → f T is

clearly linear, and when it is shown to be an injection, the theorem is proved. But if

T ∈ T (H) is such that f T = 0, then ϕ|T ψ = tr T |ψ ϕ| = f T (|ψ ϕ|) = 0 for

all ϕ, ψ ∈ H (see Theorems 3.3 (b), 3.11 (b)), so that T = 0.

Since C(H)∗ is a Banach space, we immediately obtain the following corollary:

Corollary 3.2 The space (T (H), ·

1)

is complete.

Theorem 3.13 For each S ∈ L(H) define the mapping f S : T (H) → C via the formula f S (T ) = tr ST for all T ∈ T (H). In this way we get an isometric linear bijection S → f s from the space L(H) onto the dual of the Banach space (T (H), · 1 ).

Proof From Theorem 3.8 it follows that each f S is defined on T (H) and linear.

By Theorem 3.9 | f S (T )| = |tr ST | ≤ S T 1 for all T ∈ T (H), so that f S is

continuous and f S ≤ S . The mapping S → f S is clearly linear. If ϕ, ψ ∈ H,

then by using Theorems 3.3 (b) and 3.11 (b) we see that

| ϕ|Sψ | = |tr[|Sψ ϕ|]| = | f S (|ψ ϕ|)|

≤ fS

|ψ ϕ| 1 = f S ψ

ϕ

so that

S = sup | ϕ|Sψ |

ϕ ≤ 1, ψ ≤ 1 ≤ f S .

Thus the mapping S → f S is isometric and in particular injective. We still show that it

is a surjection onto the dual of T (H). Let f ∈ T (H)∗ . We define a mapping B : H ×

H → C via the formula B(ϕ, ψ) = f (|ψ ϕ|). Then |B(ϕ, ψ)| ≤ f

|ψ ϕ| 1 =

f ϕ ψ by Theorem 3.11 (a). Since B is also conjugate linear with respect to

the first and linear with respect to the second argument, by Proposition 2.1 there is

S ∈ L(H) such that B(ϕ, ψ) = ϕ|Sψ for all ϕ, ψ ∈ H. We show that f = f S , i.e.

tr ST = f (T ) for all T ∈ T (H).

(3.11)

Both sides of (3.11) are linear functions of T that are continuous on T (H) with

respect to the norm · 1 (see Theorem 3.9). Since F(H) is dense in T (H) (see

Theorem 3.11 (e)), in view of Theorem 3.3 (c) it is enough to show that (3.11) holds

whenever T = |ψ ϕ| for some ϕ, ψ ∈ H. But according to Theorems 3.3 (b) and

3.11 (b) we get

tr S|ψ ϕ| = tr |Sψ ϕ| = ϕ|Sψ = B(ϕ, ψ) = f (|ψ ϕ|).

## Quantum measurement

## 2 The Fréchet--Riesz Theorem and Bounded Linear Operators

## 3 Strong, Weak, and Monotone Convergence of Nets of Operators

## 3 The Hilbert--Schmidt Operator Class mathcalHS(mathcalH)

## 6 The Dualities mathcalC(mathcalH)ast = mathcalT(mathcalH) and mathcalT(mathcalH)ast = mathcalL(mathcalH)

## 7 Linear Operators on Hilbert Tensor Products and the Partial Trace

## 10 A Riesz--Markov--Kakutani Type Representation Theorem for Positive Operator Measures

## 13 The Spectral Representations of Unitary and Other Normal Operators

## 2 Integration of Unbounded Functions with Respect to Positive Operator Measures

## 3 Integration of Unbounded Functions with Respect to Projection Valued Measures

## 9 Taking Stock: Hilbert Space Theory and Its Use in Quantum Mechanics

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5 Connection of the Ideals mathcalT(mathcalH) and mathcalHS(mathcalH) with the Sequence Spaces ell1 and ell2