of operators and prove that several properties from the classes Sp can be ... 2 ) ([3]). The Schatten class of operators Sp (1 ⤠p < â) is then defined. (see [3]) by.
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
CONVERGENCE IN GENERALIZED SCHATTEN. . .
� 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.
´ J.M. ALMIRA - F. PEREZ ACOSTA
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
CONVERGENCE IN GENERALIZED SCHATTEN. . .
� � 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 �
´ J.M. ALMIRA - F. PEREZ ACOSTA
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
CONVERGENCE IN GENERALIZED SCHATTEN. . .
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→∞
´ J.M. ALMIRA - F. PEREZ ACOSTA
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.
�
CONVERGENCE IN GENERALIZED SCHATTEN. . .
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)