convergence in generalized schatten classes of ...

LE MATEMATICHE Vol. LIII (1998) Fasc. II, pp. 305317

CONVERGENCE IN GENERALIZED SCHATTEN CLASSES OF OPERATORS ´ J.M. ALMIRA - F. PEREZ ACOSTA In this paper we give de�nitions for several generalized Schatten classes of operators and prove that several properties from the classes S p can be generalized to them. The main goal is characterization of convergence in these classes. We also will prove that they are quasi-Banach spaces.

1. Introduction. Let H be a Hilbert space and consider B(H, H ), the algebra of bounded linear operators T : H → H . Let T ∈ B(H, H ) be a compact operator. Then the numbers αn (T ) =

inf

sup

x1 ,...,xn−1 ∈H �x�=1,(x,x1 )=...=(x,xn−1 )=0

�T x �

and λn (T ) =

inf

rank(S) 0, �T1T2� S βp (X ) ≤

1 �T1� S βp (X ) �T2� S βp (X ) b0

�T1T2�� βp (X ) ≤

1 �T1�� βp (X ) �T2�� βp (X ) . b0

β

Proof. Let T1, T2 ∈ S p (X ). Then �T1 T2� S βp (X ) = �{bn δn (T1 T2)}∞ n=0 �l p ∞ ≤ �{bn �T1�δn (T2)}∞ n=0 �l p = �T1��{bn δn (T2)}n=0 �l p

1 �T1� S βp (X ) �T2� S βp (X ) . b0 � � β The same arguments are valid on � p (X ), � · �� βp (X ) . ≤

�

β∗

β

Corollary 2.5. Let T1 ∈ � p1 (X ) and T2 ∈ � p2 (X ), where p11 + 1 ∞ ∗ β = {bn }∞ n=0 ⊂ (0, ∞), β = { bn }n=0 and set δ = {1}n∈N . Then 1

�T1 T2�� δp (X ) ≤ 2 p �T1�� βp (X ) �T2�� β ∗ (X ) . p2

1

Proof. We have that �� ∞

2n

d (T1T2)

n=0

p

� 1p

≤

�� ∞

n

p n

d (T1) d (T2)

n=0

p

� 1p

� 1p

=

�� ∞

bnp d n (T1) p (

1 p n ) d (T2) p bn

≤

�� ∞

bnp1 d n (T1) p1

� p1 � � � p1 ∞ 1 2 1 p2 n p2 ( ) d (T2) bn n=0

n=0

n=0

= �T1�� βp (X ) �T2�� β ∗ (X ) 1

p2

1 p2

=

1 p

and

CONVERGENCE IN GENERALIZED SCHATTEN. . .

309

and, by an analogous argument, �

∞ �

d 2n+1 (T1 T2) p

n=0

� 1p

≤ �T1�� βp

1

(X ) �T2�� βp (X ) 2 ∗

also holds. Hence � �p p �T1T2�� δ (X ) ≤ 2 �T1�� βp (X ) �T2�� β ∗ (X ) p

and the proof follows.

p2

1

�

It is clear that if �{bn }∞ n=1 �l p < ∞ then ∞ ∞ �{bn d n (T )}∞ n=1 �l p ≤ �{bn �T �}n=1 �l p = �T ��{bn }n=1 �l p < ∞

and ∞ ∞ �{bn δn (T )}∞ n=1 �l p ≤ �{bn �T �}n=1 �l p = �T ��{bn }n=1 �l p < ∞

for all T ∈ B(X, Y ). Hence we have the following. β

β

Proposition 2.6. If �{bn }∞ n=1 �l p < ∞ then � p (X, Y ) = S p (X, Y ) = B(X, Y ). β

β

De�nition 2.2. We say that � p (X, Y )(S p (X, Y ), respectively) is a proper generalized Schatten Class of Operators if �{bn }∞ n=1 �l p = ∞. β

Proposition 2.7. If �{bn }∞ n=1 �l p = ∞ then � p (X, Y ) ⊆ K (X, Y ) and β S p (X, Y ) ⊆ F(X, Y ). β

/ K (X, Y ) then the sequence {d n (T )} does not Proof. Let T ∈ � p (X, Y ). If T ∈ converge to zero. Hence there exists some c > 0 such that d n (T ) ≥ c for all n. Hence ∞ �T �� βp (X,Y ) = �{bn d n (T )}∞ n=0 �l p ≥ c�{bn }n=1 �l p = ∞ β

which is in contradiction with T ∈ � p (X, Y ). The second claim has an analogous proof.

�

Proposition 2.8. For all compact operator T ∈ K (X, Y ) there exists some sequence β = {bn } ⊂ [0, ∞[ such that �{bn }∞ n=1 �l p = ∞ for all p ≥ 1 and β T ∈ � p (X, Y ).

´ J.M. ALMIRA - F. PEREZ ACOSTA

310

Proof. T ∈ K (X, Y ) implies that limn→∞ d n (T ) = 0. Hence we may choose a sequence {n k }k∈N such that n k < n k+1 and d nk (T ) < 2−k for all k. We set β = {bn }, where � 1 if ∃ k, n = n k bn = 0 otherwise. ∞ Then �{bn }n=1 �l p = ∞ for all p ≥ 1 and �T �� βp (X,Y ) = �{bn d

n

(T )}∞ n=0 �l p

=

�∞ �

nk

d (T )

p

k=0

� 1p

< ∞.

�

The same arguments are valid to prove the following. Proposition 2.9. For all T ∈ F(X, Y ) there exists some sequence β = {bn } ⊂ β [0, ∞[ such that T ∈ S p (X, Y ), �{bn }∞ n=1 �l p = ∞ for all p ≥ 1. From the fact that for any subspace X n of dimension ≤ n√of a Banach space X there exists a projection Pn : X → X on X n of norm ≤ n + 1, it can be deduced that (see [2]). Proposition 2.10. For all T ∈ B(X, Y ) and all n, δn (T ) ≤ (1 +

√

n)d n (T ).

Hence we have the following Corollary 2.11. (

bn √ ) n

S p(bn ) (X, Y ) ⊆ � p(bn ) (X, Y ) ⊆ S p 1+

(X, Y ).

�� ∞ � � � n Corollary 2.12. Let us assume that � 1+b√ � = ∞. Then n n=0 l p

� p(bn ) (X, Y ) ⊆ F(X, Y ).

Proposition 2.13. Let X be an in�nite dimensional Banach space. Let {εn } be a non-increasing sequence of positive numbers such that limn→∞ εn = 0. Then there exists an operator (with no �nite rank) T ∈ F(X, X ) such that δn (T ) ≤ εn (hence, d n (T ) ≤ εn ) for all n.

311


� Proof. We choose a sequence {an } such that ∞ k=n ak = εn for all n and let ∞ n 1 and X 0 = {0}. {x n }n=1 be a free subset of X . Set X n = span{x k }k=1 for n ≥ � 1 Denote by Pn a projection of X on X n . Then the series ∞ n=1 an �Pn � Pn is absolutely convergent, so that it converges to some operator T =

∞ �

an

n=1

1 Pn . �Pn �

Now it is clear that rank(Sn ) ≤ n for all n, where S0 = P0 and Sn =

n � k=1

ak

1 Pk �Pk �

since X 0 ⊂ . . . ⊂ X n−1 ⊂ X n ⊂ . . .. Hence δn (T ) ≤ �T − Sn | ≤

∞ �

k=n+1

ak = εn for all n.

�

Corollary 2.14. Let X be a Banach space. Then for all β = {bn } ⊂ [0, ∞) β β there are T ∈ � p (X, X ) and S ∈ S p (X, X ), operators with no �nite rank. A normed linear space Y has the extension property (cf. [2]) if for all normed linear space X and all M linear subspace of X and T ∈ B(M, Y ), there exists an extension T ∈ B(X, Y ) with �T � = �T �. It is well known that if Y has the extension property then all T ∈ B(X, Y ) satisfy the relation d n (T ) = δn (T ) for all n. Hence we have the following β

β

Proposition 2.15. If Y has the extension property then � p (X, Y ) = S p (X, Y ) for all X and β .

3. Convergence in generalized Schatten classes of operators. In what follows we will assume that {bn } ⊂ (0, ∞) and max{b2n , b2n+1 } sup bn n∈N �

�

= C < ∞,

to be sure hat the generalized Schatten classes are quasinormed spaces.


312

Theorem 3.1. The sequence of operators {Tm } converges to T in the topolβ ogy � only if �Tm − T � → 0 and the family of series � of S p (X ) if and p ∞ is equiconvergent. �{bn δn (Tm )}n=0 �l p m∈N

Proof. (⇒) 1 �Tm b0

�Tm − T � S βp (X ) → 0 implies �Tm − T � → 0, since �Tm − T � ≤

− T � S βp (X ) .

1

Let ε > 0. Then there exists some n 0 ∈ N such that �Tm − T � S βp (X ) < ε p for all m ≥ n 0 . On the other hand �Tm − T � S βp (X ) ≤ ∞ for all m < n 0 . Hence there exists some k0 ∈ N such that   1p ∞ � 1 p  bk δk (Tm − T ) p  < ε p for all m ∈ N k=k0

and

 

But ∞ ��

p

∞ �

k=k0

b2k (Tm ) p

k=k0

p

 1p

1

bk δk (T ) p  < ε p .

� 1p

∞ ∞ � � � � 1 1 p p ≤C ( bk δk (Tm − T ) p ) p + ( bk δk (T ) p ) p k=k0

k=k0

1

and

≤ 2Cε p ∞ ��

p

b2k+1 (Tm ) p

k=k0

� 1p

∞ ∞ � � � � 1 p p p 1p ≤C ( bk δk (Tm − T ) ) + ( bk δk (T ) p ) p k=k0

k=k0

1 p

Hence

≤ 2Cε . ∞ �

k=2k0

p

bk δk (Tm ) p ≤ (2C) p ε

313


� � p and the family of series �{bn δn (Tm )}∞ n=0 �l p

m∈N

is equiconvergent.

β

(⇐) Let {Tm } ∪ {T }�⊂ S p (X ) such that �Tm − T � → 0 and the family of series � p is equiconvergent, and let ε > 0. Then there exists �{bn δn (Tm )}∞ n=0 �l p m∈N some k0 ∈ N such that ∞ �

k=k0

p

bk δk (Tm ) p ≤ ε

for all m ∈ N

and ∞ �

k=k0

p

bk δk (T ) p ≤ ε.

Hence �� ∞

p b2k δ2k (Tm

k=k0

≤C

�� ∞

− T)

� 1p

p

p bk δk (Tm ) p

k=k0

� 1p

+

∞ ��

1 p bk δk (T ) p ) p

k=k0

�

1

≤ 2Cε p and �� ∞

p b2k+1 δ2k+1 (Tm

k=k0

≤C

�� ∞

− T)

p bk δk (Tm ) p

k=k0

� 1p

p

� 1p

+

∞ ��

p bk δk (T ) p

k=k0

1 p

≤ 2Cε . Hence ∞ �

k=2k0

p

bk δk (Tm − T ) p ≤ (2C) p ε

� 1p �


314 and

�Tm −

p T � S β (X ) p

≤

� 2k� 0 −1

≤

� 2k� 0 −1

p bk

k=0

k=0

=

2k 0 −1 �

p bk δk (Tm

k=0

p

− T) +

∞ �

k=2k0

p

bk δk (Tm − T ) p

� ∞ � p p bk δk (Tm − T ) p �Tm − T � + k=2k0

� p bk �Tm − T � p + (2C) p ε

which approaches to zero since �Tm − T � → 0.

�

Theorem 3.2. The sequence of operators {Tm } converges to T in the topoβ logy of � p (X ) if and � � only if �Tm − T � → 0 and the family of series p

�{bn d n (Tm )}∞ n=0 �l p

m∈N

is equiconvergent.

Proof. The proof follows the same arguments as in Theorem 3.1.

�

β

Proposition 3.3. Let T ∈ S p (X ) and set S(t) = exp(t T ) − I for all t ≥ 0. β Then S(t) ∈ Sp (X ) and lim �S(t) − Sm (t)� S βp (X ) = 0

m→∞

where Sm (t) =

m � tkT k k=1

k!

.

� p� Proof. �S(t) − Sm (t)� → 0 is clear. To prove that �{bn δn (Sm (t))}∞ n=0 �l P m∈N is equiconvergent, we note that if rank(R) ≤ n with R ∈ B(X, X ) then

315


T R ∈ B(X, X ) and rank(T R) ≤ n for any T ∈ B(X, X ). Hence δn (Sm (t)) ≤

m m �� t k k � t k k−1 � � � T − T R� � rank(R)≤n k! k!

inf

k=1

k=1

m � �� t k k−1 � � T (T − R)� ≤ inf � rank(R)≤n k! k=1

≤ =

m k � k=1

t �T �k−1 inf �T − R� rank(R)≤n k!

�� m k=1

� tk k−1 �T � δn (T ) k!

≤ tet �T � δn (T ) and

∞ �

k=k0

p

bk δk (Sm (t)) p ≤ t p et �T � p

∞ �

p

bk δk (T ) p

k=k0

β S p (X ).

and the equiconvergence follows, since T ∈ β To �nalize the proof we must show that S(t) ∈ S p (X ). To do this, we observe that the inequality |δn (Sm (t)) − δn (S(t))| ≤ �Sm (t) − S(t)� implies δn (S(t)) = lim δn (Sm (t)) ≤ sup δn (Sm (t)) ≤ tet �T � δn (T ) m→∞

m∈N β

β

for all n ∈ N and now it is clear that T ∈ S p (X ) implies S(t) ∈ Sp (X ). β

�

β

Theorem 3.4. (� p (X ), � · �� βp (X ) ) and (S p (X ), � · � S βp (X ) ) are complete. β

Proof. Let {Tm } be a Cauchy sequence in S p (X ) and suppose (without loss of generality) that supm∈N �Tm � S βp (X ) ≤ M < ∞. Then {Tm } is also Cauchy in B(X, X ) because of the inequality �Tm 1 − Tm 2 � ≤ b10 �Tm 1 − Tm 2 � S βp (X ) . Since B(X, X ) is complete, there exists an operator T ∈ B(X, X ) such that lim �T − Tm � = 0.

m→∞


316

Let N ∈ N and ε > 0 be arbitrarily chosen. Then there exists some n = n(ε, N ) ∈ N such that � �� N p bk �T − Tn � p < ε. k=0

Hence N � k=0

p

b2k δ2k (T ) p ≤ C p ≤2

N � k=0

p−1

C

p

N �

p bk δk (T

k=0

≤ 2 p−1C p ≤2

p

bk (δk (T − Tn ) + δk (Tn )) p

p−1

p

�� N k=0

p

− Tn ) + 2

p−1

C

p

N �

p

bk δk (Tn ) p

k=0

� N � p p bk �T − Tn � p + 2 p−1 C p bk δk (Tn ) p k=0

p

C (ε + M ).

On the other hand N � k=0

p

b2k+1 δ2k+1 (T ) p ≤ 2 p−1C p (ε + M p )

has an analogous proof. It follows that 2N+1 � k=0

p

bk δk (T ) p ≤ 2 p C p (ε + M p )

for all N and ε. Hence �T � S βp (X ) ≤ 2C M β

and T ∈ S p (X ). Let ε > 0. Then there exists an m 0 ∈ N such that �Tk − Tm � S βp (X ) ≤ ε for all k, m ≥ m 0 . Using the same arguments as above and noticing that T − Tm = T − Tk + Tk − Tm , it is clear that for all m ≥ m 0 , �T − Tm � S βp (X ) ≤ 2Cε. β

β

This proves that S p (X ) is complete. The proof for � p (X ) is analogous.

�


317

Acknowledgments. The authors thanks the referee for his valuable comments and suggestions (specially for his remarks to Proposition 2.13, Corollary 2.14 and Proposition 3.3).

REFERENCES [1] [2] [3]

A. Pietsch, Eigenvalues and s-numbers, Cambridge University Press, 1987. A. Pinkus, n-Widths in Approximation Theory, Springer-Verlag, 1985. K.H. Zhu, Operator Theory in Function Spaces, Marcel-Dekker, 1990.

Department of Mathematical Analysis, La Laguna University, 38271 La Laguna, Tenerife (SPAIN)