ON THE RANGE AND THE KERNEL
Pn
OF THE ELEMENTARY OPERATORS
i=1 Ai XBi − X
S. MECHERI
Abstract. Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex
Hilbert space H into itself. For A = (A1 , A2 ...A
n ) and B = (B1 , B2 ...Bn ) n-tuples in B(H), we define the elementary
P
operator ∆A,B X : B(H) 7→ B(H) by ∆A,B =
Ai XBi − X. In this paper we show that if ∆A,B = 0 = ∆∗A,B , then
T + ∆A,B (X) ≥ kT k
I
I
for all X ∈ I (proper bilateral ideal) and for all T ∈ ker(∆A,B | I).
1.
Introduction
Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of operators
on H into itself. Given A, B ∈ B(H), we define the generalized derivation δA,B :P
B(H) 7→ B(H) by δAB (X) =
n
AX − XB, and the elementary operator ∆AB : B(H) 7→ B(H) by ∆A,B (X) = i=1 Ai XBi − X, where A =
(A1 , A2 ...An ) and B = (B1 , B2 ...Bn ) are n-tuples in B(H). Note δA,A = δA , ∆A,A = ∆A . Let
B(H) ⊃ K(H) ⊃ Cp ⊃ F (H)(0 < p < ∞)
denote, respectively, the class of all bounded linear operators, the class of compact operators, the Schatten p−
class, and the class of finite rank operators on H. All operators herein are assumed to be linear and bounded.
Received January 3, 2003.
2000 Mathematics Subject Classification. Primary 47B47, 47A30, 47B20; Secondary 47B10.
Key words and phrases. elementary operators, Schatten p-classes, unitairy invariant norm, orthogonality.
This Work was Supported by the Research Center Project No. Math/1422/10.
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Let k.kp , k.k∞ denote, respectively, the Cp -norm and the K(H)-norm. Let I be a proper bilateral ideal of B(H).
It is well known that if I 6= {0}, then K(H) ⊃ I ⊃ F (H).
In [1, Theorem 1.7], J. Anderson shows that if A is normal and commutes with T , then for all X ∈ B(H),
(1.1)
kT + δA (X)k ≥ kT k .
Over the years, Anderson’s result has been generalized in various ways. Some results concern elementary
operators on B(H) such as X → AXB − X or δA,B (X) = AX − XB; since these are not normal derivations, some
extra condition is needed in each case to obtain the orthogonality result. In [4], P. B. Duggal established the
orthogonality result for ∆AB under the hypothesis that (A, B) satisfies a generalized Putnam-Fuglede property
(which is one way to generalize normality).
Another way to generalize Anderson’s result is to consider the restriction of an elementary operator (e.g.,
X → AXB − X , δA,B (X) = AX − XB or ∆AB ) to a norm ideal (I, kkI ) of B(H). Among the results in this
direction, Duggal [6] has obtained the orthogonality result for ∆AB | Cp (the restriction to the Schatten p-class
Cp ) under the Putnam-Fuglede hypothesis on (A, B), and F. Kittaneh [8], [9] proved the orthogonality result for
restricted generalized derivations δA,B | I (with the Putnam-Fuglede condition for (A, B)).
In [16], A. Turnsek initiated a different approach to generalize Anderson’s theorem, one which does not rely
on the normality via the Putnam-Fuglede condition.
Turnsek [16, Theorem 1.1] proved that if φ is a contractive map on a (fairly
Pn general normed algebra A, then
φ(s) = s implies kφ(x) − x + sk ≥ ksk for every x in A. Let φ(X) =
i=1 Ai XBi ; thus, if kφk ≤ 1, then
∆AB (S) = 0 implies k∆AB (X) − Sk ≥ kSk for every operator X in B(H), i.e., the range and the kernel of ∆AB
are orthogonal [16,
1.2]. Turnsek also obtained an analogue of the orthogonality
result
Pn Proposition
Pn
Pnfor ∆AB∗ | Cp .
∗
∗
∗
∗
Let
∆
(X)
=
A
XB
−
X.
Turnsek’s
result
[16,Theorem
2.4]
is
that
if
A
A
≤
1,
i
i
i=1 i i
i=1 Ai Ai ≤ 1,
Pn AB∗
Pi=1
n
∗
∗
B
B
≤
1,
B
B
≤
1,
then
for
S
∈
C
,
∆
(S)
=
∆
(S)
=
0
implies
that
k∆
(X)
−
Skp ≥ kSkp .
p
AB
AB
AB
i=1 i i
i=1 i i
The main result of this note is a direct extension of Turnesek’s theorem from Cp to a general norm ideal I. Other
related results are also given.
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2.
Prelimenaries
Let T ∈ B(H) be compact, and let s1 (X) ≥ s2 (X) ≥ . . . ≥ 0 denote the singular values of T , i.e., the eigenvalues
1
of |T | = (T ∗ T ) 2 arranged in their decreasing order. The operator T is said to be belong to the Schatten p-class
Cp if
"∞
# p1
X
1
kT kp =
sj (T )p
= [tr(T )p ] p ,
1 ≤ p < ∞,
i=1
where tr denotes the trace functional. Hence C1 is the trace class, C2 is the Hilbert-Schmidt class, and C∞ is the
class of compact operators with
kT k∞ = s1 (T ) = sup kT f k
kf k=1
denoting the usual operator norm. For the general theory of the Schatten p-classes the reader is referred to [7],
[14], [15].
Note that the Ky Fan norm kT kn is defined by
kT kn =
n
X
sj (T )
j=1
for n ≥ 1.
Each unitarily invariant norm k.kI satisfies
kU AkI = kAV kI
for all unitaries U and V (provided that kAkI < ∞), and is defined on a natural subclass Ik.kI of B(H), called
the norm ideal associated with k.kI . Whereas the (unitarily invariant) usual norm k.k is defined on all of B(H),
other invariant norms are defined on norm ideals contained in the ideal K(H) of compact operators in B(H), see
[7].
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Definition 2.1. let C be complex numbers and let E be a normed linear space. Let x, y ∈ E, if kx − λyk ≥
kλyk for all λ ∈ C, then x is said to be orthogonal to y. Let F and G be two subspaces in E. If kx + yk ≥ kyk,
for all x ∈ F and for all y ∈ G, then F is said to be orthogonal to G.
3.
Main results
Our main results are the following
3.1.
I be a bilateral ideal of B(H) and C = (C1 , C2 , . . . , Cn ) n-tuple of operators in B(H). If
PnTheorem
PLet
n
∗
∗
∗
C
C
≤
1,
C
i=1 i i
i=1 i Ci ≤ 1 and ∆c (T ) = 0 = ∆c (T ), then
kT + ∆c (X)kI ≥ kT kI
(3.1)
for all X ∈ I and for all T ∈ ker ∆c ∩ I.
Proof. By virtue of [7, p. 82], it suffices to show that for all n ≥ 1,
k(∆c (X)) + T kn =
n
X
sj (∆c (X)) + T ) ≥
j=1
n
X
sj (T ) = kT kn .
j=1
Let T = U |T | be the polar decomposition of T where U is a partial isometry and ker U = ker |T |. Then for all
j ≥ 1 the result of Gohberg and Krein [7, p. 27] guarentees that
sj (∆c (X) + T ) ≥ sj (U ∗ [∆c (X) + T ]) = sj (U ∗ (∆c (X)) + |T |).
Recall that if {gn }n≥1 is an orthonormal basis of H, then it results from [7, p. 47] that for all n ≥ 1,
n
X
j=1
sj (U ∗ (∆c (X)) + |T |) ≥
n
X
|h[U ∗ (∆c (X)) + |T |]gj , gj i| .
j=1
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Consequently we get,
(3.2)
n
X
sj (∆c (X) + T ) ≥
j=1
n
X
|h[U ∗ (∆c (X)) + |T |]gj , gj i| =
X
.
j=1
Pn
Pn
It is known that if i=1 Ci Ci∗ ≤ 1, i=1 Ci∗ Ci ≤ 1 and ∆c (T ) = 0 = ∆∗c (T ) then the eigenspaces corresponding
2
to distinct non-zero eigenvalues of the compact positive operator |T | reduce each Ci , see ([3, Theorem 8], [16,
Lemma 2.3]). In particular |T | commutes with Ci for all 1 ≤ i ≤ n. Hence
Ci |T | = |T | Ci .
This shows the existence of an orthonormal basis {eki } ∪ {fm } of H such that {fm } is an orthonormal basis of
ker |T | and {eki } consists of common eigenvectors of Ci and |T |. If
{gn } = {eki } ∪ {fm } ,
since for all m ≥ 1,
U fm = |T | fm = 0,
(3.2) becomes
X
=
n
X
j=1
n
∗
X
[U (∆C (X))]ekj , ekj .
|T | ekj , ekj +
j=1
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Therefore for all n ≥ 1,
n
X
inf(n, card(eki ))
sj ((∆C (X) + T )) ≥
j=1
X
h|T | ej , ej i
j=1
inf(n, card(eki ))
=
X
sj (T ) ≥
j=1
n
X
sj (T ) = kT kn .
j=1
Theorem 3.2. Let I be a bilateral
of B(H) P
and A = (A1 , A2P
, . . . , An ), B = P
(A1 , A2 , . . . , An ) nPideal
n
n
n
n
∗
∗
∗
∗
tuples of operators in B(H). If
A
A
≤
1,
A
A
≤
1,
B
B
≤
1,
i=1 i i
i=1 i i
i=1 i i
i=1 Bi Bi ≤ 1 and
∗
ker ∆A,B ⊆ ker ∆A,B , then
(3.3)
kS + ∆A,B (X)kI ≥ kSkI ,
for all X ∈ I and for all S ∈ ker(∆c | I).
Proof. It suffices to take the Hilbert space H ⊕ H, and operators
Ai 0
0 T
Ci =
,
S=
,
0
Bi
0 0
X=
0 X
0 0
and apply Theorem 3.1 and use the fact that the norm of a matrix is greater than or equal to the norm of an
entry along the main diagonal of the matrix [7].
Corollary 3.1. Let I be a bilateral
and A = (A1 , A2P
, . . . , An ), B = (A
Pnideal of∗ B(H) P
Pn1 , A2 ,∗. . . , An ) nn
n
∗
∗
tuples of operators in B(H). If
A
A
≤
1,
A
A
≤
1,
B
B
≤
1,
i=1 i i
i=1 i i
i=1 i i
i=1 Bi Bi ≤ 1 and
ker ∆A,B ⊆ ker ∆∗A,B (S), then ker(∆nA,B | I) = ker(∆A,B | I).
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Proof. The result of S. Bouali and S. Cherki [2] guarentees that
R(An ) = ker(A)
if, and only if,
R(A) ∩ ker A = {0} ,
where A ∈ B(E) and E is a complex vector space. In particular
R(∆nA,B | I) = ker(∆A,B | I)
if and only if,
R(∆A,B | I) ∩ ker(∆A,B | I) = {0}
which holds from the above theorem.
4.
(1)
A Comment and some open questions
It is well known that the Hilbert-Schmidt class C2 is a Hilbert space under the inner product
hY, Zi = trZ ∗ Y.
We remark here that for the Hilbert Schmidt norm k.k2 , the orthogonality results in Theorem 3.2 is to be
understood in the usual Hilbert space sense. Note in the case where I = C2 , then
2
2
2
kT + ∆A,B (X)k2 = k∆A,B (X)k2 + kT k2 ,
if and only if ker ∆A,B ⊆ ker ∆∗A,B , for all X ∈ C2 and for all T ∈ ker ∆A,B ∩ I. This can be seen as an immediate
consequence of the fact that
R(∆A,B | C2 )⊥ = ker(∆A,B | C2 )∗ = ker(∆∗A,B | C2 ),
(2) If the assumptions of Theorem 3.2 holds, then ran∆A,B ∩ ker ∆AB = {0}, where the closure can be taken
in the most weak (uniform) norm. Hence ∆A,B (∆A,B (X)) = 0 implies ∆A,B (X) = 0.
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Indeed if Z ∈ ran∆A,B ∩ ker ∆A,B , then Z = lim ∆A,B (Xn ) and ∆A,B (Z) = 0. By applying Theorem 3.2 we
n−→∞
get
k∆A,B (Xn ) − ZkI ≥ kZkI
so,
kZ − ZkI ≥ kZkI .
Then Z = 0. We deduce that R(∆A,B ) ⊕ ker ∆AB = I.
(3) I s the sufficient condition in Thorem 3.2 necessary?
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
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S. Mecheri, King Saud University College of Science, Department of Mathematics, P.O.Box 2455, Riyadh 11451, Saudi Arabia, e-mail:
mecherisalah@hotmail.com
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