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

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

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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







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|).



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