Applied Mathematical Sciences, Vol. 6, 2012, no. 136, 6777 - 6786
Isolated Points of Spectrum for
Quasi - ∗ - Class A Operators
A. Sekar
Department of Mathematics
Sri Ramakrishna Engineering College
Vattamalaipalayam, Coimbatore - 641 022, Tamil Nadu, India
[email protected]
C. V. Seshaiah
Department of Mathematics
Sri Ramakrishna Engineering College
Vattamalaipalayam, Coimbatore - 641 022, Tamil Nadu, India
[email protected]
D. Senthil Kumar
Post Graduate and Research Department of Mathematics
Government Arts College (Autonomous)
Coimbatore - 641 018,Tamil Nadu, India
[email protected]
P. Maheswari Naik
Department of Mathematics
Sri Ramakrishna Engineering College
Vattamalaipalayam, Coimbatore - 641 022, Tamil Nadu, India
[email protected]
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A. Sekar, C. V. Seshaiah, D. Senthil Kumar and P. Maheswari Naik
Abstract. Let T be a Quasi - ∗ - class A operator on a complex Hilbert
space H if T ∗ (|T 2| − |T ∗ |2 )T ≥ 0. In this paper, we prove that if E is the Riesz
idempotent for a non-zero isolated point λ of the spectrum of T ∈ Quasi - ∗ class A operator, then E is self-adjoint and EH = ker(T − λ) = ker (T − λ)∗ .
We will also prove a necessary and sufficient condition for T ⊗ S to be quasi ∗ - class A where T and S are both non-zero operators.
Mathematics Subject Classification: Primary 47A10; Secondary 47B20
Keywords: ∗ - paranormal operators, Weyl’s theorem, ∗ - class A operators,
Quasi - ∗ - class A operators, generalized a - Weyl’s theorem, B - Fredholm, B
- Weyl
1. Introduction
Let B(H) denote the algebra of all bounded linear operators acting on an
infinite dimensional separable Hilbert space H. For a positive operators A
and B, write A ≥ B if A − B ≥ 0. If A and B are invertible and positive
operators, it is well known that A ≥ B implies that log A ≥ log B. However
[2], log A ≥log B does not necessarily imply A ≥ B. A result due to Ando
[5] states that for invertible positive operators A and B, log A ≥ log B if and
r
r
1
only if Ar ≥ (A 2 B r A 2 ) 2 for all r ≥ 0. For an operator T , let U|T | denote the
polar decomposition of T , where U is a partially isometric operator, |T | is a
positive square root of T ∗ T and ker (T ) = ker (U) = ker (|T |), where ker (S)
denotes the kernel of operator S.
An operator T ∈ B(H) is positive, T ≥ 0, if (T x, x) ≥ 0 for all x ∈ H,
and posinormal if there exists a positive λ ∈ B(H) such that T T ∗ = T ∗ λT .
Here λ is called interrupter of T . In other words, an operator T is called
posinormal if T T ∗ ≤ c2 T ∗ T , where T ∗ is the adjoint of T and c > 0 [7]. An
operator T is said to be heminormal if T is hyponormal and T ∗ T commutes
with T T ∗ . An operator T is said to be p - posinormal if (T T ∗ )p ≤ c2 (T ∗ T )p
for some c > 0. It is clear that 1 - posinormal is posinormal. An operator
T is said to be p - hyponormal, for p ∈ (0, 1), if (T ∗ T )p ≥ (T T ∗ )p . An 1 -
Isolated points of spectrum
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hyponormal operator is hyponormal which has been studied by many authors
and it is known that hyponormal operators have many interesting properties
similar to those of normal operators [19]. Furuta et al [9], have characterized
class A operator as follows. An operator T belongs to class A if and only if
1
(T ∗ |T |T ) 2 ≥ T ∗ T .
An operator T is said to be paranormal if ||T 2x|| ≥ ||T x||2 and ∗ - paranormal if ||T 2x|| ≥ ||T ∗ x||2 for all unit vector x ∈ H. Recently, B. P. Duggal
et al [8] have considered the new class of operators : An operator T ∈ B(H)
belongs to ∗ - class A if |T 2| ≥ |T ∗ |2 . The authors of [14] have extended ∗ class A operators to quasi - ∗ - class A operators. An operator T ∈ B(H) is
said to be quasi - ∗ - class A if T ∗ |T 2|T ≥ T ∗ |T ∗ |2 T and quasi - ∗ - paranormal
if ||T ∗T x||2 ≤ ||T 3 x||||T x|| for all x ∈ H. An operator T is said to be Quasi - ∗
- class A [12] operator on a complex Hilbert space H if T ∗ (|T 2 | − |T ∗ |2 )T ≥ 0.
As a further generalization, Mecheri [13] has introduced the class of k - quasi
- ∗ - class A operators. An operator T is said to be k - quasi - ∗ - class A
operator on a complex Hilbert space H if T ∗k (|T 2 | − |T ∗2 )T k ≥ 0 where k is a
natural number.
An operator T is called normal if T ∗ T = T T ∗ and (p, k) - quasihyponork
mal if T ∗ ((T ∗ T )p − (T T ∗)p )T k ≥ 0 (0 < p ≤ 1, k ∈ N). A. Aluthge [1],
B.C. Gupta [6], S.C. Arora and P. Arora [3] introduced p - hyponormal, p quasihyponormal and k - quasihyponormal operators, respectively.
p - hyponormal ⊂ p - posinormal ⊂ (p, k) - quasiposinormal,
p - hyponormal ⊂ p -quasihyponormal ⊂
(p, k) - quasihyponormal ⊂ (p, k) - quasiposinormal
and
hyponormal ⊂ k - quasihyponormal ⊂ (p, k) - quasihyponormal
⊂ (p, k) - quasiposinormal
for a positive integer k and a positive number 0 < p ≤ 1.
If T ∈ B(H), we shall write N(T ) and R(T ) for the null space and the
range of T , respectively. Also, let σ(T ) and σa (T ) denote the spectrum and
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A. Sekar, C. V. Seshaiah, D. Senthil Kumar and P. Maheswari Naik
the approximate point spectrum of T , respectively. An operator T is called
Fredholm if R(T ) is closed, α(T ) = dim N(T ) < ∞ and β(T ) = dim H/R(T ) <
∞. Moreover if i(T ) = α(T ) − β(T ) = 0, then T is called Weyl. The essential
spectrum σe (T ) and the Weyl σW (T ) are defined by
σe (T ) = {λ ∈ C : T − λ is not Fredholm}
and
σW (T ) = {λ ∈ C : T − λ is not Weyl},
respectively. It is known that σe (T ) ⊂ σW (T ) ⊂ σe (T )∪ acc σ(T ) where we
write acc K for the set of all accumulation points of K ⊂ C. If we write iso
K = K\ acc K, then we let
π00 (T ) = {λ ∈ iso σ(T ) : 0 < α(T − λ) < ∞}.
We say that Weyl’s theorem holds for T if
σ(T )\σW (T ) = π00 (T ).
Let σp (T ) denotes the point spectrum of T , i.e., the set of its eigenvalues.
Let σjp (T ) denotes the joint point spectrum of T . We note that λ ∈ σjp (T ) if
and only if there exists a non-zero vector x such that T x = λx , T ∗ x = λx.
It is evident that σjp (T ) ⊂ σp (T ). It is well known that, if T is normal,
then σjp (T ) = σp (T ). If T = U|T | is the polar decomposition of T and
λ = |λ|eiθ be the complex number, |λ| > 0, |eiθ | = 1. Then λ ∈ σjp (T )
if and only if there exist a non-zero vector x such that Ux = eiθ , |T |x =
|λ|x. Let σap (T ) denotes the approximate point spectrum of T , i.e., the set
of all complex numbers λ which satisfy the following condition: there exists
a sequence {xn } of unit vectors in H such that limn (T − λ)xn = 0. It
is evident that σp (T ) ⊂ σap (T ). Let σjap (T ) be the joint approximate point
spectrum of T , then λ ∈ σjap (T ) if and only if there exists a sequence {xn } of
unit vectors such that limn→∞ (T − λ)xn = limn→∞ (T ∗ − λ)xn = 0. It is
evident that σjap (T ) ⊂ σap (T ) for all T ∈ B(H). It is well known that, for a
normal operator T , σjap (T ) = σap (T ) = σ(T ).
An operator T ∈ B(H) is said to have the single-valued extension property
(or SVEP) if for every open subset G of C and any analytic function f : G → H
such that (T −z)f (z) ≡ 0 on G, we have f (z) ≡ 0 on G. An operator T ∈ B(H)
is said to have Bishop’s property (β) if for every open subset G of C and every
sequence fn : G → H of H - valued analytic functions such that (T − z)fn (z)
converges uniformly to 0 in norm on compact subsets of G, fn (z) converges
Isolated points of spectrum
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uniformly to 0 in norm on compact subsets of G. An operator T ∈ B(H) is
said to have Dunford’s property (C) if HT (F ) is closed for each closed subset
F of C. It is well known that
Bishop’s property (β) ⇒ Dunford’s property (C) ⇒ SVEP.
Let T ∈ B(H) and let λ0 be an isolated point of σ(T ). Then there exists a
positive number r > 0 such that {λ ∈ C : |λ − λ0 | ≤ r} σ(T ) = {λ0 }. Let γ
be the boundary of {λ ∈ C : |λ − λ0 | ≤ r}. Then
1
(λ − T )−1 dλ,
E = 2πi
γ
is called the Riesz idempotent of T for λ0 . Then it is well known that
E 2 = E, ET = T E, σ(T |ranE ) = {λ0 } and ker(T − λ0 I) ⊆ ranE.
In general, it is well known that the Riesz idempotent E is not an orthogonal
projection and a necessary and sufficient condition for E to be orthogonal is
that E is self-adjoint. In [15], Stampfli showed that if T satisfies the growth
condition G1 , then E is self-adjoint and E(H) = ker(T − λ0 ). Recently, Jeon
and Kim [11] and Uchiyama [18] obtained Stampfli’s result for quasi - class
A operators and paranormal operators. In general even if T is a paranormal
operator, the Riesz idempotent E of T with respect to λ0 is not necessarily
self - adjoint. We show that if E is the Riesz idempotent for a nonzero isolated
point λ0 of the spectrum of a quasi - ∗ - class A operator T , then E is self adjoint and EH = ker(T − λ0 ) = ker(T ∗ − λ0 ).
2. Main Results
Jun Li and et al [12] have introduced quasi - ∗ - class A operators and have
proved many interesting properties of it.
Lemma 2.1. ([12, Theorem 2.2, Theorem 2.3]) (1) Let T ∈ B(H) be quasi ∗ - class A operator and T does not have a dense
range, then
A B
T =
0 0
where A = T |ranT is the restriction of T to ranT and A ∈ ∗ - class A operator.
Moreover σ(T ) = σ(T1 ) ∪ {0}.
(2) If T is an quasi - ∗ - class A operator and M is its invariant subspace,
then the restriction T |M of T to M is also an quasi - ∗ - class A operator.
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A. Sekar, C. V. Seshaiah, D. Senthil Kumar and P. Maheswari Naik
Lemma 2.2. [12, Theorem 2.4] Let T ∈ B(H) is an quasi - ∗ - class A
operator. If λ = 0 and (T − λ)x = 0 for some x ∈ H, then (T − λ)∗ x = 0.
Lemma 2.3. Let T ∈ B(H) is an quasi - ∗ - class A operator. Then T is
isoloid.
Proof. Let T ∈ B(H) is an quasi - ∗ - class A operator with representation given
in Lemma 2.1. Let z be an isolated point in σ(T ). Since σ(T ) = σ(T1 ) ∪ {0},
z is an isolated point in σ(T1 ) or z = 0. If z isolated point in σ(T1 ), then
/ σ(T1 ). Then for x ∈ ker(T3 ),
z ∈ σp (T1 ). Assume that z = 0 and z ∈
−T1−1 T2 x ⊕ x ∈ kerT . This completes the proof.
Theorem 2.4. Let A ∈ B(H) is an quasi - ∗ - class A operator and let λ be
a non-zero isolated point of σ(A). Let Dλ denote the closed disk that centered
at λ such that Dλ σ(A) = {λ}. Then the Riesz idempotent
1
(λ − A)−1 dλ
E = 2Πi
∂Dλ
satisfies
EH = ker(A − λ) = ker((T − λ)∗ ).
In particular, E is self adjoint.
Proof. If A is quasi - ∗ - class A operator, then λ is an eigenvalue of A and
EH = ker(A − λ) by Lemma 2.3. Since ker(A − λ) ⊂ ker(A − λ)∗ by Lemma
2.2, it suffices to show that ker(A − λ)∗ ⊂ ker(A − λ) . Since ker(A − λ) is
a reducing subspace of A by Lemma 2.2 and the restriction of a quasi - ∗ class A operator to its reducing subspaces is also a quasi - ∗ - class A operator
by Lemma 2.1, hence A can be written as follows: A = λ ⊕ A1 on H =
ker(A − λ) ⊕ (ker(A − λ))⊥ , where A1 is ∗-class A with ker(A1 − λ) = {0}.
Since
λ ∈ σ(A) = {λ} ∪ σ(A1 )
is isolated, the only two cases occur, one is λ ∈ σ(A1 ) and the other is that λ is
an isolated point of σ(A1 ) and this contradicts the fact that ker(A1 −λ) = {0}.
Since A1 is invertible as an operator on (ker(A−λ))⊥ , ker(A−λ) = ker(A−λ)∗ .
Next, we show that E is self-adjoint. Since
EH = ker(A − λ) = ker(A − λ)∗ ,
Isolated points of spectrum
6783
we have ((z − A)∗ )−1 E = (z − λ)−1 E. Therefore
1
∗
((z − A)∗ )−1 Edz
E E=−
2πi ∂D
1
(z − A)−1 Edz
=−
2πi ∂D
1
(z − A)−1 dz)E
=(
2πi ∂D
= E.
This completes the proof.
3. Tensor product of quasi - ∗ - class A operators
The tensor products T ⊗ S preserves many properties of T, S ∈ B(H), but
by no means all of them. Thus, whereas the normaloid property is invariant
under tensor products; again, whereas T ⊗ S is normal if and only if T and S
are normal [10, 16], there exist paranormal operators T and S such that T ⊗ S
is not paranormal [4]. It is shown in [11] that T ⊗ S is quasi-class A if and only
if S, T are quasi-class A operators. In the following theorem we will prove a
necessary and sufficient condition for T ⊗ S to be quasi - ∗ - class A operator
where T and S are both non-zero operators.
Recall that (T ⊗ S)∗ (T ⊗ S) = T ∗ T ⊗ S ∗ S and so, by the uniqueness of
positive square roots, |T ⊗ S|r = |T |r ⊗ |S|r for any positive rational number r.
From the density of the rationales in the real, we obtain |T ⊗ S|p = |T |p ⊗ |S|p
for any positive real number p. If T1 ≥ T2 and S1 ≥ S2 , then T1 ⊗ S1 ≥ T2 ⊗ S2
(see, [17])
Theorem 3.1. Let S, T ∈ B(H) be non-zero operators. Then T ⊗ S is quasi
- ∗ - class A operator if and only if one of the following holds:
a) S and T are quasi - ∗ - class A operators.
b) S 2 = 0 or T 2 = 0.
Proof. Since T ⊗ S is quasi - ∗ - class A operator if and only if (T ⊗ S)∗ (|(T ⊗
S)|2 − (|(T ⊗ S)∗ |))(T ⊗ S) ≥ 0
⇔ T ∗ (|T 2| − |T ∗ |2 )T ⊗ S ∗ |S 2 |S + T ∗ |T ∗ |2 T ⊗ S ∗ (|S 2 | − |S ∗ |2 )S ≥ 0.
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A. Sekar, C. V. Seshaiah, D. Senthil Kumar and P. Maheswari Naik
Hence the sufficiency is clear. Conversely, assume that T ⊗ S is quasi - ∗ class A operator. Then for every x, y ∈ H we have
(3.1)
T ∗ (|T 2 | − |T ∗ |2 )T x, xS ∗ |S 2 |Sy, y + T ∗ |T ∗ |2 T x, xS ∗ (|S 2 | − |S ∗ |2 )Sy, y ≥ 0
It suffices to prove that if (a) does not hold, then (b) holds. Suppose that
2
T = 0 and S 2 = 0.
To the contrary, assume that T is not a quasi - ∗ - class A operator, then
there exists x0 ∈ H such that
T ∗ (|T 2 | − |T ∗ |2 )T x0 , x0 = α < 0 and T ∗ |T ∗ |2 T x0 , x0 = β > 0.
From (3.1) we have
(3.2)
α + βS ∗ |S 2 |Sy, y ≥ βS ∗|S ∗ |2 Sy, y
for all y ∈ H.
Thus S is quasi - ∗ - class A operator since α + β ≤ β. Using the HölderMcCarthy inequality we have
1
1
S ∗ |S 2 |Sy, y = (S ∗2 S 2 ) 2 Sy, Sy ≤ ||Sy||2(1− 2 ) S ∗2 S 2 Sy, Sy1/2 = ||Sy||||S 3y||
and
S ∗ |S ∗|2 Sy, y = SS ∗ Sy, Sy = (S ∗S)y, S ∗Sy
= ||S ∗Sy||2.
Thus
(3.3)
(α + β)||Sy||||S 3y|| ≥ β||S ∗Sy||2.
Since S is a quasi - ∗ - class A operator, Lemma 2.1 imply that
S1 S2
S=
on H = ran(S k ) ⊕ ker S ∗k .
0 0
Then S1 is ∗ - class A, S3k = 0 and σ(S) = σ(S1 ) ∪ {0}. Therefore (3.3) implies
(α + β)||S1η||||S13η|| ≥ β||S1∗S1 η||2
for all η ∈ ran S k . Since S1 is ∗ - class A and ∗ - class A is normaloid. Thus
taking supremum on both sides of the above inequality, we have
(α + β)||S1 ||4 ≥ β||S1∗ S1 ||2 .
Isolated points of spectrum
Therefore S1 = 0. Since
S2 =
0 S2
0
6785
2
0
= 0.
Hence S k+1 = 0. This contradicts the assumption S 2 = 0. Hence T must be a
quasi - ∗ - class A operator. A similar argument shows that S is also quasi ∗ - class A operator. This completes the proof.
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Received: September, 2012