Far East Journal of Mathematical Sciences (FJMS) Volume …, Number …, 2011, Pages … Available online at http://pphmj.com/journals/fjms.htm Published by Pushpa Publishing House, Allahabad, INDIA
SPECTRAL PROPERTIES OF CLASS A COMPOSITION OPERATORS D. Senthilkumar, T. Prasad and S. M. Sherin Joy Department of Mathematics Government Arts College (Autonomous) Coimbatore-641018, Tamilnadu, India e-mail:
[email protected] [email protected] Abstract In this paper, some spectral properties of class A composition operators on L2 space are studied by using semi-hyponormal composition operator transform and quasi class A composition operators are characterized.
1. Introduction and Preliminaries Let ( X , Σ, λ ) be a sigma finite measure space. If w is a non-negative complex valued Σ measurable function, then the weighted composition operator W on L2 ( X , Σ, λ ) induced by a non-singular measurable
transformation T from X into itself with the condition that the measure λT −1 is absolutely continuous with respect to the measure λ and the RadonNikodym derivative d λT −1 dλ = f 0 is essentially bounded is given by 2011 Pushpa Publishing House 2010 Mathematics Subject Classification: 47B20, 47B38. Keywords and phrases: semi-hyponormal operators, class A operators, quasi class A operators, weighted composition operators. Received August 25, 2011
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D. Senthilkumar, T. Prasad and S. M. Sherin Joy
Wf = w( f T ) , f ∈ L2 ( X , Σ, λ ). In case that w = 1, we say that W is a composition operator denoted by
C. The Radon-Nikodym derivative of the measure λ(T k )−1 with respect to λ is denoted by f 0( k ) , where T k is obtained by composing T − k times.
Every essentially bounded complex valued measurable function f 0 induces the bounded operator M f 0 on L2 (λ ) , which is defined by M f 0 f = f 0 f for every f ∈ L2 (λ ). To examine the weighted composition operators effectively, Lambert [14] associated conditional expectation operator E with T as E ( ⋅ T −1Σ ) = E (⋅). E ( f ) is defined for each non-negative measurable function f ∈
Lp (1 p ) and is uniquely determined by the conditions (i) E ( f ) is T −1Σ measurable. (ii) If B is any T −1Σ measurable set for which we have
∫ B fdλ
converges, then
∫ B fdλ = ∫ B E ( f ) dλ.
As an operator on L p , E is the projection on to the closure of range of C. E is the identity on L p if and only if T −1Σ = Σ. Detailed discussion of E is found in [5, 9, 11]. Let f be a complex valued measurable function on X.
The essential range of f denoted by ER( f ) is given by
ER ( f ) = {λ : λ ∈ and λ f −1 ( F ) ≠ 0 for every neighborhood F of λ}. Let B( H ) denote the algebra of all bounded linear operators on complex Hilbert space H. Then an operator T ∈ B( H ) is said to be p-hyponormal, 0 < p ≤ 1, if (T ∗T ) p ≥ (TT ∗ ) p . If p = 1, then T is hyponormal and if p=
1 , then T is semi-hyponormal. In [18], Xia studied semi-hyponormal 2
Spectral Properties of Class A Composition Operators
3
operators extensively. The class of p-hyponormal operators was first studied in [1] by Aluthge. An operator T ∈ B( H ) is class A if T 2 ≥ T 2 . Class A operators has been introduced and studied by Furuta et al. [10]. Recently in [13], Jeon and Kim introduced quasi class A operators. An operator T ∈ B( H ) is quasi class A if T ∗ T 2 T ≥ T ∗ T 2 T . The inclusion relations
among these classes are as follows: p-hyponormal (0 < p < 1) ⊆ class A ⊆ quasi class A. See [10] and [13].
The Aluthge transform, an operator transform to convert an operator to another being close to normal operator which shares some spectral properties ~ ~ with the first one, of T ∈ B( H ) is the operator T given by T = 1
1 2
T 2U T
was introduced in [1] by Aluthge. More generally, we may
form the family of operators {Tr : 0 < r ≤ 1}, where Tr = T r U T 1− r [2]. For a composition operator C, the polar decomposition is given by 1 C = U C , where C f = f 0 f and Uf = f T . In [3], Lambert f0 T has given more general Aluthge transformation for composition operators as r r
Cr = C U C
1− r
f0 2 and Cr f = f T . That is Cr is weighted f 0 T r
f 2 composition operator with weight π = 0 , where 0 < r < 1. f0 T Spectra of composition operators have been investigated by mathematicians [4, 16, 17]. Our main aim of this paper is to study results on the spectrum of class A composition operators via its hyponormal composition operator transformation and characterize class A composition operators.
2. Main Results
many some semiquasi
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D. Senthilkumar, T. Prasad and S. M. Sherin Joy Let σ(C ) and σ jp (C ) denote the spectrum and joint point spectrum of
C ∈ L2 (λ ) , respectively. If C denotes the set of complex numbers, then a point z ∈ C is in the joint point spectrum σ jp (C ) if there exists a non-zero vector x such that Cx = zx and C ∗ x = z x. The following proposition due to Campbell and Jamison [5] is well known:
Proposition 2.1. For w ≥ 0, (i) W ∗Wf = f 0 [E ( w 2 )] T −1 f . (ii) WW ∗ f = w( f 0 T ) E ( wf ) .
Theorem 2.2. Let C be class A operator. If
Σ −1 = Σ , then
12 f f 0 T . ∈ ER( f 0 E ( w 2 ) T −1 )}, where w = f0 T 1 2 0
σ(C ) ⊂ {z ∈ C : z 2
Proof. If C ∈ B( L2 (λ )) is class A, then the operator Cˆ = C U C is its semi-hyponormal operator transform [8]. For f ∈ L2 ,
Cˆ f = C U C f =
1 1 1 f 0 f 02 T f 02 f 02 T ⋅ f T = ⋅ f T. f0 T f0 T
12 f f 0 T . Thus Cˆ is weighted composition operator with weight w = f0 T 1 2 0
ˆ = Since Cˆ is weighted composition operator by Proposition 2.1, Cˆ *Cf
Spectral Properties of Class A Composition Operators
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f 0 E ( w 2 ) T −1 f . So σ ( Cˆ *Cˆ ) = ER ( f 0 E ( w2 ) T −1 ) . Since Cˆ is semihyponormal composition operator, if x ∈ ER( f 0 E ( w 2 ) T −1 ) , then there exists a z ∈ σ( Cˆ ) such that z
2
= x [7]. Hence 2
σ (C ) ⊂ { z ∈ : z ∈ ER ( f 0 E ( w2 ) T −1 )}. Since Σ −1 = Σ , Cˆ is semi-hyponormal if and only if Cˆ is hyponormal [6]. Thus, σ(C ) = σ(Cˆ ) by [8, Theorem 2.2]. Hence, 2
σ ( C ) ⊂ { z ∈ : z ∈ ER ( f 0 E ( w2 ) T −1 )}.
Theorem 2.3. Let C be class A operator. If Σ −1 = Σ , then 2
σ p ( C ) ⊂ { z ∈ : z = f 0 E ( w 2 ) T −1 } . Proof. Since C is class A, Cˆ = C U C
is semi- hyponormal. If
z ∈ σ jp (CˆT ) , then there exists a non-zero f such that Cˆ *Cˆ = z
2
f . Since
ˆ = f E ( w2 ) T −1 f Cˆ is semi-hyponormal, σ jp (Cˆ ) = σ p (Cˆ ). Since Cˆ *Cf 0 for
f ∈ L2 (λ ) ,
and
since
σ p (C ) = σ p (Cˆ )
[8],
2
σ p ( C ) ⊂ { z ∈ : z = f 0 E ( w 2 ) T −1 } .
it
follows
that
Since σ p (C ) has symmetry about 0 except on the unit circle [16], the following corollary is immediate:
Corollary 2.4. Let C ∈ B( L2 (λ )) is class A and 1 ∉ ER( f 0 E ( w 2 ) T −1 ). Then σ p (C ) ⊆ σ p (C ∗ ) . If C is compact, then σ p (C ) − {0} is a finite subset of unit circle [16]. The following theorem due to Panayappan and Senthilkumar [15] sharpens the results:
Theorem 2.5. Let C be compact. Then σ(C ) = σ p (C ) .
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D. Senthilkumar, T. Prasad and S. M. Sherin Joy
Theorem 2.6. Let L2 (λ ) be infinite dimensional. Then no class A composition operators on L2 (λ ) is compact.
Proof. Let C be a compact class A composition operator on L2 (λ ). Then Cˆ = C U C is semi-hyponormal and compact and so Cˆ must be normal. Since Cˆ is compact, σ(Cˆ ) = σ p (Cˆ ) by theorem 2.5 . Since σ p (Cˆ ) is contained in the unit circle, σ(Cˆ ) is in the unit circle. But σ p (Cˆ ) = σ(C )
[8]. Thus we have C is unitary, a contradiction.
The Weyl’s spectrum of a composition operator C on L2 (λ ) is the set W (C ) = {λ : C − λ ∉ F0 }, where F0 is the set of all Fredholm operators of index zero.
Theorem 2.7. Let C be class A with 1 does not belong to ER( f 0 E ( w 2 ) T −1 ) . Then σ(C ) = W (C ) .
Proof. Since C is class A, Weyl’s theorem holds for C. Then W (C ) = σ(C ) − π00 (C ) , where π00 (C ) is the set of isolated eigenvalues of finite multiplicity of C. It is enough to show that π00 (C ) is empty. If 0 ∈ π00 (T ) , then C has closed range and so C is invertible, which contradicts the fact that C is not invertible. Let λ ≠ 0 is an eigenvalue of C with
λ ≠ 1. Since σ p (C ) has
symmetry about 0 except on the unit circle [16], λ cannot be isolated, λ ∉ π00 (C ). This completes the proof.
Now we characterize quasi class A compositions as follows:
Theorem 2.8. C ∈ B( L2 (λ )) is quasi class A if and only if J11 2 f ≥ J1 2 f
for all f ∈ L2 (λ ) , where J1 = f 0( 2 ) [ E (( f 0( 2 ) )1 2 ) T −1 ] and J =
f 0 [E ( f 0 ) T −1 ].
Spectral Properties of Class A Composition Operators
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Proof. Let C be quasi class A. Then the following inequality holds: C ∗ C 2 C ≥ C ∗ C 2 C. Thus C 2 1 2C 2 f , f ≥
C C 2 f, f
and so C2 1 2C f 2 ≥ Since C 2 1 2 C = ( M ( 2 ) )1 4 C , f
C C | f 2.
(2.1)
C 2 1 2 C is a weighted composition
0
operator with weight ( f 0( 2 ) )1 4 . Since
C 2 1 2 C is a weighted composition operator with weight
( f 0( 2 ) )1 4 , its polar decomposition is given by C 2 1 2 C = VP, where P = C2 1 2C = M J 1
and
1 4 χ supp (σ ( J 1 )) Vf = ( f 0(2 ) ) f T, J1
where
(2) J1 = f 0( 2 ) E f 0 2 T −1 and supp(σ( J1 )) denotes the support of σ( J1 ) . 1
Also,
C C =M
f 01 2
C. Thus
C C is a weighted composition operator
with weight f 01 2 . Therefore,
χ f 01 2 σ ( J ) σ (J )
C C = V1P1 , where
P1 = C C = M
J
and V1 f =
f T , where J = f 0 [E ( f 0 ) T −1 ].
Then by (2.1), C is quasi class A if and only if M J f 2 ≥ M J f 2 1 and so C is quasi class A if and only if J11 2 f ≥ J 1 2 f , , f ∈ L2 ( λ ) ,
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D. Senthilkumar, T. Prasad and S. M. Sherin Joy
where J1 = f 0( 2 ) [E (( f 0( 2 ) )1 2 ) T −1 ] and J = f 0 [E ( f 0 ) T −1 ].
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