10 A. L. CAREY, V. GAYRAL, A. RENNIE, and F. A. SUKOCHEV

in norm, it also converges in measure, and so we may apply the Fatou Lemma,

[26, Theorem 3.5 (i)], to deduce that

τ

(

(1 +

D2)−p/4−1/4nT ∗T

(1 +

D2)−p/4−1/4n

)

≤ lim inf

k→∞

τ

(

(1 +

D2)−p/4−1/4nTk ∗Tk(1

+

D2)−p/4−1/4n

)

.

Since the same conclusion holds for TT

∗

in place of T

∗T

, we see that

Qn(T ) ≤ lim inf

k→∞

Qn(Tk) = lim

k→∞

Qn(Tk) ∞,

and so T ∈ B2(D,p). Finally, fix ε 0 and n ≥ 1. Now choose N large enough so

that Qn(Tk − Tl) ≤ ε for all k, l N. Applying the Fatou Lemma to the sequence

(Tk)k≥1, we have Qn(T − Tl) ≤ lim infk→∞ Qn(Tk − Tl) ≤ ε. Hence Tk → T in the

topology of B2(D,p).

We now give some easy but useful stability properties of the algebras B2(D,p).

Lemma 1.7. Let T ∈ B2(D,p), S ∈ N and let f ∈ L∞(R).

(1) The operators Tf(D), f(D)T are in B2(D,p). If moreover T ∗ = T , then

Tf(T ) ∈ B2(D,p). In all these cases,

Qn(Tf(D)), Qn(f(D)T ), Qn(Tf(T )) ≤ f

∞

Qn(T ).

(2) If

S∗S

≤ T

∗T

and

SS∗

≤ TT

∗,

then S ∈ B2(D,p) with Qn(S) ≤ Qn(T ).

(3) We have S ∈ B2(D,p) if and only if |S|,

|S∗|

∈ B2(D,p).

(4) The real and imaginary parts (T ), (T ) belong to B2(D,p).

(5) If T = T

∗,

let T = T+ − T− be the Jordan decomposition of T into pos-

itive and negative parts. Then T+, T− ∈ B2(D,p). Consequently B2(D,p) =

span{B2(D,p)+}.

Proof. (1) Since T (1 +

D2)−s/4,

T

∗(1

+

D2)−s/4

∈

L2(N

, τ), we immediately

see that

Tf(D)(1 +

D2)−s/4

= T (1 +

D2)−s/4f(D),

¯(D)T

f

∗(1

+

D2)−s/4

∈

L2(N

, τ),

and when T is self-adjoint, we also have

Tf(T )(1 +

D2)−s/4

= f(T )T (1 +

D2)−s/4,

¯(T

f )T (1 +

D2)−s/4

∈

L2(N

, τ).

To prove the inequality we use the trace property to see that

τ((1 +

D2)−s/4

¯(D)T

f

∗Tf(D)(1

+

D2)−s/4)

= τ(T (1 +

D2)−s/4|f|2(D)(1

+

D2)−s/4T ∗)

≤ f

2

∞

τ((1 +

D2)−s/4T ∗T

(1 +

D2)−s/4),

and similarly for Tf(D) and Tf(T ) when T ∗ = T .

(2) Clearly, ϕs(S∗S) ≤ ϕs(T ∗T ) and ϕs(SS∗) ≤ ϕs(TT ∗). The assertion follows

immediately.

(3) This follows from Qn(T ) = (Qn(|T |) + Qn(|T

∗|))/2.

Item (4) follows since

B2(D,p) is a ∗-algebra, and then item (5) follows from (2), since for a self-adjoint

element T ∈ B2(D,p):

T

∗T

=

|T|2

= (T+ +

T−)2

= T+

2

+ T−

2

≥ T+,

2 T−.2

This completes the proof.