Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 603041, 6 pages
doi:10.1155/2009/603041
Research Article
A Note on the Range of the Operator
X → T X − XT Defined on C2 H
Vasile Lauric
Department of Mathematics, Florida A&M University, Tallahassee, FL 32307, USA
Correspondence should be addressed to Vasile Lauric, vasile.lauric@famu.edu
Received 10 November 2008; Accepted 11 May 2009
Recommended by Manfred H. Moller
We show how a proof of J. Stampfli can be extended to prove that the operator X → T X − XT
defined on the Hilbert-Schmidt class, when T is an M-hyponormal, p-hyponormal, or loghyponormal operator, has a closed range if and only if σT is finite.
Copyright q 2009 Vasile Lauric. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
Let H be a complex, separable, infinite dimensional Hilbert space, and let LH denote the
algebra of all linear bounded operators on H. The Hilbert-Schmidt class, denoted by C2 H, is
a Hilbert space with the ||·||2 -norm that arises from the inner product X, Y trXY ∗ , where
tr is the scalar-valued trace. For T ∈ LH, define ΔT : LH → LH by ΔT X T X − XT,
and let σT denote the spectrum of T. Let the range of a linear operator S be denoted by
RS. For a normal operator N ∈ LH, Anderson and Foiaş 1 proved that RΔN is norm
closed if and only if σN is a finite set. In 2, Stampfli extended this result to the class of
hyponormal operators.
Theorem A 2. Let T ∈ LH be a hyponormal operator. Then RΔT is norm closed if and only
if σT is finite.
In fact, Stampfli provided a proof of the “only if” implication which extends to a
larger class of operators than the class of hyponormal operators see Proposition 2.2. For
an operator T ∈ LH, let σnap T denote its normal approximate point spectrum, that is, the set
of those λ ∈ C for which there exists an orthonormal sequence {φn }n in H such that
T − λφn T − λ∗ φn −→ 0.
1.1
2
International Journal of Mathematics and Mathematical Sciences
Define the class GH as follows:
GH : T ∈ LH | σnap T is an infinite set .
1.2
Some classes of hyponormal related operators, such as M-hyponormal operators, that is,
m · T − λ∗ φ ≤ T − λφ,
∀φ ∈ H, ∀λ ∈ C, for some m > 0,
1.3
p-hyponormal operators, that is, T ∗ T p ≥ T T ∗ p for some p > 0, or log-hyponormal
operators, that is, invertible operators such that logT ∗ T ≥ logT T ∗ , have spectrum that
is finite or they belong to GH. Particularly, the hyponormal operators i.e., 1-hyponormal
have this property.
In 3 Stampfli proved the following lemma which will be used in Section 2.
Lemma B. Let T ∈ GH and let {λn }∞
n1 be a sequence of distinct points of σnap T . Then for
of
positive
numbers
converging to zero, there exists an orthonormal sequence
any sequence {εn }∞
n1
of
vectors
in
H
such
that
{φn }∞
n1
T − λn φn T − λn ∗ φn < εn for n 1, 2, . . . ,
φn , T φk 0 for k 1, . . . , n − 1.
1.4
1.5
2
2. The Closedness of the Range of ΔT
The operator ΔT defined on the Hilbert-Schmidt class will be denoted in the remainder of this
2
2
2
note by ΔT , that is, ΔT : C2 H → C2 H, ΔT X T X − XT. Let H M H denote the set
of M-hyponormal operators.
2
Proposition 2.1. Let T ∈ H M H. If σT is finite, then RΔT is closed.
Proof. It is well known that an operator T ∈ H M H with finite spectrum is normal. Indeed,
for such an operator, the restriction to an invariant subspace M belongs to H M M. On the
other hand, if T ∈ H M H with σT {λ}, then T λI, cf. 4. Thus, we can write
0
λi Ei , where Ei ’s are the spectral projections.
T ni1
2
Let Xn and C be in C2 H such that ΔT Xn − C2 → 0. Therefore ΔT Xn −C → 0 in
the LH norm, and according to Theorem A, there exists X 0 ∈ LH such that C T X 0 −X 0 T.
For an arbitrary X ∈ LH, let Xij be the block-matrix representation of X relative to the
0
⊕Ei H. Thus
decomposition H ni1
Cij λi − λj Xij0 ,
2.1
for all i, j 1, . . . , n0 . This implies that each Xij0 1/λi −λj Cij is a Hilbert-Schmidt operator.
Moreover Xii0 can be chosen 0, and thus X 0 ∈ C2 H.
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3
2
Proposition 2.2. Let T ∈ GH. Then RΔT is not closed.
Proof. We will use same notation and circle of ideas as in 2. Let {λn }n≥1 be sequence of
distinct points of σnap T so that λn → λ0 . Let
−1/2
ηn max λj1 − λj | j 1, . . . , n ,
2.2
and choose a nonincreasing sequence {εn }n≥1 so that 0 < εn ≤ |λn1 − λn |2 , n ≥ 1, and
2 2
n≥1 εn ηn < ∞. According to Lemma B, there exists an orthonormal sequence {φn }n ≥1 that
satisfies 1.4 and1.5. Let H1 ∨{φn | n ≥ 1}, H2 H⊥1 , and let δn such that
T φn μn φn δn ,
δn ⊥ φn ,
n ≥ 1.
2.3
δn < 2εn ,
n ≥ 1.
2.4
It results that
μn − λn < εn ,
Define V : H → H by V φn |λj1 − λj |−1/2 φn1 , n ≥ 1, and V g 0, g ∈ H2 . Let Mn ∨{φj |
j 1, . . . , n} and let Pn be the orthogonal projection onto Mn , and define Vn V Pn . A tedious
calculation shows that
ΔT Vn φj ⎧ ⎨vj μj1 − μj φj1 vj δj1 − Vn δj ,
j ≤ n,
⎩−V δ ,
n j
j > n,
2.5
where vj |λj1 − λj |−1/2 . Denoting ΔT Vn − ΔT Vm by Δn,m
T , then for n < m,
⎧
⎪
0,
⎪
⎪
⎨
Δn,m
T φj ⎪−vj μj1 − μj φj1 vj δj1 Vm − Vn δj ,
⎪
⎪
⎩
Vm − Vn δj ,
j ≤ n,
n < j ≤ m,
2.6
j > m.
Furthermore, from 2.3 it results that
δj ⊥ φj , φj1 , φj2 , . . .
2.7
and from 2.4
Vn δj ≤ 2ηj εj ,
∀j, n ≥ 1.
2.8
We will show next that Δn,m
T 2 → 0 when m, n → ∞, thus there exists C ∈ C2 H
2
such that ΔT Vn − C2 → 0, that is, C ∈ RΔT .
4
International Journal of Mathematics and Mathematical Sciences
First, we will show that Δn,m
T |H1 2 → 0, when m, n → ∞. Indeed,
2
∞ n,m 2 Δ | Δn,m φj 2 2.6
H1 2
T
T
j1
m −vj μj1 − μj φj1 vj δj1 Vm − Vn δj 2
jn1
2.9
∞ Vm − Vn δj 2 .
jm1
The first sum of the right-hand side of the above can be majorized by
2·
m m −vj μj1 − μj φj1 vj δj1 2 2 ·
Vm − Vn δj 2 .
jn1
2.10
jn1
Since φj1 ⊥ δj1 , we have
⎤
⎡
m ∞ n,m 2
2
2 2
2
2
Δ | ≤ 2⎣
Vm − Vn δj ⎦.
vj μj1 − μj vj δj1 H1 2
T
jn1
2.11
jn1
According to 2.8,
Vm − Vn δj 2 ≤ 16η2 ε2 ,
j j
2.12
and according to 2.4,
2
2
vj2 δj1 ≤ 4ηj2 εj1
≤ 4ηj2 εj2 ,
μj1 − μj 2 ≤ 2εj λj1 − λj 2 ≤ 8ε2 2λj1 − λj 2 ,
j
2.13
which implies
2
vj2 μj1 − μj ≤ 8ηj2 εj2 2λj1 − λj .
2.14
∞
m n,m 2
2 2
Δ | ≤ c1 ·
λj1 − λj ,
η
ε
c
·
2
H
j j
T
1 2
2.15
Therefore
jn1
jn1
where c1 and c2 are some constants. After a careful review of the proof, one can see that the
sequence {λn } can be assumed to converge fast enough otherwise choose a subsequence of
it, more precisely m
jn1 |λj1 − λj | → 0 when n, m → ∞.
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5
We show next that Δn,m
T |H2 2 → 0, when m, n → ∞. Indeed, we can write
2
T ∗ φn μn φn γn
with γn , φn 0,
γn ≤ 2εn ,
n ≥ 1.
2.16
Obviously, we can write T ∗ φn θn φn γn with γn , φn 0, which implies
θn θn φn γn , φn T ∗ φn , φn φn , T φn φn , μn φn δn μn ,
1.4,2.4
∗
γn T − μ φn ≤ T − λn ∗ φn λn − μ ≤ 2εn .
n
n
2.17
For an orthonormal basis {ψi }i≥1 of H2 , we will show that
∞ Δn,m ψi 2 −→ 0
T
when n, m −→ ∞.
2.18
i1
For each i, write T ψi ∞
k1
i
ak φk wi with wi ∈ H2 . Thus
Vm T ψi m
m
i
i
ak Vm φk Vm wi ak vk φk1 .
k1
k1
2.19
Since Vm ψi 0, we have ΔT Vm ψi −Vm T ψi , and consequently, for n < m,
Δn,m
T ψi m
i
ak vk φk1 .
2.20
kn1
Since the sequence {φk } is orthonormal, we have
m n,m 2
i 2 2
Δ ψi · vk .
a
T
k 2.21
kn1
Therefore
m ∞ ∞ m
∞ i 2 2
i 2
2
Δn,m ψi 2 vk
ak · vk ak .
T
i1
i1 kn1
kn1
i1
2.22
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International Journal of Mathematics and Mathematical Sciences
For a fixed k,
∞ ∞
∞ 2 i 2 ψi , T ∗ φk 2
T ψi , φk ak i1
i1
2.16
i1
∞ i1
∞ ψi , μ φk γk 2 ψi , γk 2 ≤ γk 2
k
2.23
i1
2.16
≤ 4εk2 .
Consequently,
∞
i1
Δn,m
T ψi ≤ 4
2
The operator C is not in
C/
∈ RΔT .
m
kn1
2
RΔT vk2 · εk2 → 0 for n, m → ∞.
since, according to the proof of Theorem A in 2,
2
Theorem 2.3. Let T ∈ H M H. Then RΔT is closed if and only if σT is finite.
2
Proof. If T ∈ H M H and σT are finite, then according to Proposition 2.1, RΔT is
closed. Conversely, if T ∈ H M H has an infinite spectrum, then there are infinitely many
distinct points {λn }n that are either isolated points of the spectrum, in which case they
are eigenvalues, or accumulation points of the spectrum, in which case they are in σap T .
Since T ∈ H M H, we have σp T , σap T ⊆ σnap T . Thus T ∈ GH and according to
2
Proposition 2.2, RΔT is not closed.
References
1 J. Anderson and C. Foiaş, “Properties which normal operators share with normal derivations and
related operators,” Pacific Journal of Mathematics, vol. 61, no. 2, pp. 313–325, 1975.
2 J. G. Stampfli, “On the range of a hyponormal derivation,” Proceedings of the American Mathematical
Society, vol. 52, pp. 117–120, 1975.
3 J. G. Stampfli, “Compact perturbations, normal eigenvalues and a problem of Salinas,” Journal of the
London Mathematical Society, vol. 9, no. 1, pp. 165–175, 1974.
4 A. Uchiyama and T. Yoshino, “Weyl’s theorem for p-hyponormal or M-hyponormal operators,”
Glasgow Mathematical Journal, vol. 43, no. 3, pp. 375–381, 2001.