Composition operators of (α,β)- normal operators

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Composition operators of (α,β)- normal operators D.SenthilKumar, SherinJoy.S.M., Post Graduate and Research Department of Mathematics, Govt. Arts College, Coimbatore- 641 018, Tamilnadu, India. email : [email protected] Abstract: Here in this paper, composition operators and weighted composition operators of (α,β)normal operators on L2 space and composition operators of (α,β)- normal operators on general weighted Hardy Spaces are characterised. Keywords and Phrases: Hyponormal composition operators, m -hyponormal composition operators, (α,β)- normal operators. 2010 Mathematics subject classification: 47B33, 47B20. 1. Preliminaries Let (X, Σ, λ) be a sigma-finite measure space and T be a non-singular measurable transformation from X onto itself. Composition transformation C on L2 (λ) induced T is given by Cf = f ◦ T for every f in L2 (λ). If C is bounded, we call C a composition operator on L2 (λ). It is known that T induces a bounded composition operator C on L2 (λ) if and only if the measure λ T −1 is absolutely continuous with respect the the measure λ and f0 is essentially bounded where f0 is the Radon-Nikodym derivative of the measure λ T −1 with respect to λ. The Radon-Nikodym derivative of the measure λ(T k )−1 with respect to λ is denoted by f0k , where T k is obtained by composing T , k times [8]. Every essentially bounded complex-valued measurable function f0 induces the bounded operator Mf0 on L2 (λ), which is defined by Mf0 f = f0 f , for every f ∈ L2 (λ) and it is well known that C ∗ C = Mf0 . A weighted composition operator W (w.c.o) induced by T is a linear transformation acting on the set of complex valued Σ measurable functions f of the form W f = wf ◦ T , where w is a complex valued Σ measurable function. When w = 1, we say that W is a composition operator. Let wk denote w(w ◦ T )(w ◦ T 2 )....(w ◦ T k−1 ) so that W k f = wk (f ◦ T )k [11]. For better examination, Lambert [9] associated with each transformation T , a condition expectation operator E(•/T −1 Σ) = E(•) and has been studied in [3],[5],[6]. E(f ) is defined for each non-negative measurable function f , or for each f ∈ Lp (1 ≤ p) and is uniquely determined by the conditions (i) E(f ) is T −1 Σ is measurable. R R (ii) If B is any T −1 Σ measurable set for which B f dλ converges, we have B E(f )dλ. 2. (α,β)- normal composition operators on L2 space Let B(H) be the Banach Algebra of all the bounded linear operators on a Hilbert Space H. An operator T ∈ B(H) is said to be hyponormal if T T ∗ ≤ T ∗ T , and m-hyponormal if there exists an m ≥ 0 such that T T ∗ ≤ m2 T ∗ T . Correspondingly the composition operator C is hyponormal if and only if (f0 ◦T )P ≤ f0 a.e and m-hyponormal if there exists an m ≥ 0 such that (f0 ◦T )P ≤ m2 f0 a.e.[11] Definition 2.1[9] An operator T ∈ B(H) is said to be an (α,β)- normal operator [10], (0≤ α ≤ 1 ≤ β) if α2 T ∗ T ≤ T T ∗ ≤ β 2 T ∗ T . When α = β = 1, (α,β)- normal operator is normal. We need the following lemma, due to Harrington and Whitley [8] for the characterization of (α,β)- normal composition operators on L2 space.

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Lemma 2.2 Let P denote the projection of L2 onto R(C). Then (a) C ∗ Cf = f0 f and CC ∗ f = (f0 ◦ T )P f , for all f ∈ L2 . (b) R(C) = { f ∈ L2 : T −1 (Σ) is measurable }. Theorem 2.3 Let C ∈ B(L2 (λ)), then C is (α,β)- normal if and only if α2 f0 ≤ (f0 ◦ T )P ≤ β 2 f0 a.e. Proof: By definition 2.1, C is (α,β)- normal if and only if α2 C ∗ C ≤ CC ∗ ≤ β 2 C ∗ C for (0≤ α ≤ 1 ≤ β). Since C ∗ C = Mf0 and CC ∗ f = Mf0 ◦T P , it follows that C is (α,β)- normal if and only if α2 Mf0 ≤ Mf0 ◦T P ≤ β 2 Mf0 a.e. Thus α2 f0 ≤ (f0 ◦ T )P ≤ β 2 f0 a.e for 0≤ α ≤ 1 ≤ β. Corollary 2.4 Let C ∈ B(L2 (λ)).If C has a dense range, then C is (α,β)- normal if and only if α2 f0 ≤ (f0 ◦ T ) ≤ β 2 f0 a.e. Corollary 2.5 Let C ∈ B(L2 (λ)). If C has a dense range, then C ∗ is (α,β)- normal if and only if α2 (f0 ◦T ) ≤ f0 ≤ β (f0 ◦ T ) a.e. Example 2.6 2

Let X = N , the set of all natural numbers and λ be the counting measure on it. Define T : N → N by T (1) = 1, T (2) = 1, T (5n p + m − 2) = n + 1 for m = 0,1,2...and n ∈ N . Then C is (α,β)- normal for 0 ≤ α ≤ 1 and β ≥ 5/3. Now we give an example of an m - hyponormal composition operator on L2 (λ) which is neither (α,β)- normal nor hyponormal. Example 2.7 Let X = N , the set of all natural numbers and λ be the counting measure on it. Define T : N → N by T (1) = 2, T (2) = 1, T (3) = 2, T (3n + m) = n + 2 for m = 1,2...and n ∈ N . Then C is m -hyponormal but neither (α,β)- normal nor hyponormal for n = 1. Now we use the following proposition due to Campbell and Jamison[3] for the characterization of weighted (α,β)- normal composition operators on L2 space. Proposition 2.8 For w ≥ 0 (a). W ∗ W f = f0 [E(w2 )] ◦ T −1 f . (b). W W ∗ f = w(f0 ◦ T )E(wf ). Here we characterise weighted (α,β)- normal composition operators. Theorem 2.9 If T −1 Σ = Σ, then W is (α,β)- normal if and only if α2 f0 (w2 ) ◦ T −1 ≤ w2 (f0 ◦ T ) ≤ β f0 (w2 ) ◦ T −1 a.e. Proof: 2

(k)

(k)

Since W k f = wk (f ◦ T k ) and (W ∗k )f√= f0 E(wk f ) ◦ T −k , we have W ∗k W k = f0 E(wk2 ) ◦ −k T f and |W ∗ |f = vE(vf ) where v= w.√ f0 ◦T 2 1 . If T −1 Σ = Σ then E becomes the identity [E(w f0 ◦T ) ] 4

operator and hence W W ∗ f = v 4 f = w2 (f0 ◦ T )f ; f ∈ L2 . If W is (α,β)- normal, then α2 W ∗ W ≤ W W ∗ ≤ β 2 W ∗ W and hence α2 f0 (w2 ) ◦ T −1 ≤ w2 (f0 ◦ T ) ≤ β 2 f0 (w2 ) ◦ T −1 a.e.

3 1 1 The Aluthge transformation[1] of T is the operator T˜ is given by T˜ =| T | 2 U | T | 2 . r 1−r More generally, we may form the family of operators Ar : 0 < r ≤ 1 where Ar =|A| U |A| √ . For a composition operator C, the polar decomposition is given by C=U |C| where |C|f = f0 f and U F = √f1◦T f ◦ T . Lambert et al.[5] has given general Aluthge transformation for composition op0

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0 erators as Cr =|C|r U |C|1−r and Cr f =( f0f◦T ) 2 f ◦ T . That is, Cr is weighted composition operator 1 2

0 with weight π=( f0f◦T ) where 0 < r < 1. Since Cr is a weighted composition operator it is easy to √ √ show that |Cr |f = f0 [E(π)2 ◦ T −1 ]f and |Cr∗ |f = vE[vf ] where v= π.√ f0 ◦T 2 1 . Also we have

[E(π f0 ◦T ) ] 4

Crk f =πk (f ◦ T k ). (k) Cr∗k f =f0 E(πk f ) ◦ T −k . (k) Cr∗k Crk f =f0 E(πk2 ) ◦ T −k f . Corollary 2.10. If T −1 Σ = Σ, then Cr is (α,β)- normal if and only if α2 f0 (π 2 ) ◦ T −1 ≤ π 2 (f0 ◦ T ) ≤ β f0 (π ) ◦ T −1 a.e. 2

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1 The second Aluthge transformation of T described by B.P.Duggal [4] is given by T˜ =| T˜ | 2 1 V | T˜ | 2 , where T˜ = V | T˜ | is the polar decomposition of T˜. 1 1 SenthilKumar and Prasad [14] studied that the operator C˜ =| Cr | 2 V | Cr | 2 where Cr = V | Cr | is the polar decomposition of the generalised Aluthge transformation Cr : 0 < r < 1, 1 ◦ T ) where J = f0 E(π 2 ) ◦ T −1 . is weighted composition operator with weight w0 = J 4 π( χsupJ 1 J4 Corollary 2.11.

If T −1 Σ = Σ, then W is (α,β)- normal if and only if α2 f0 (w02 ) ◦ T −1 ≤ w02 (f0 ◦ T ) ≤ β f0 (w02 ) ◦ T −1 a.e. 2

3. (α,β)- normal Composition Operators on Weighted Hardy spaces P∞ n H 2 (γ) of formal complex power series f (z) = such that kf k2γ = n=0 an Z P∞ The2 set 2 < ∞ is the general Hardy space of functions analytic in the unit disc with the |a | γ n n=0 n inner product P∞ hf, giγ = n=0 an bn γn2 P∞ n for f as above and g(z) = n=0 bn Z and γ = {γn }∞ n=0 be a sequence of positive numbers with → 1 as n → ∞. γ0 = 1 and γn+1 γn If φ is an analytic function mapping the unit disc D into itself, we define the composition operator Cφ on the spaces H 2 (γ) by Cφ f = f0 φ Though the operator Cφ are defined everywhere on the classical Hardy space H 2 (the case when γn = 1 for all n), they are not necessarily defined on all of H 2 (β). The composition operator Cφ is defined on H 2 (γ) only when the function φ is analytic on some open set containing the closed unit disc having supremum norm strictly smaller than one [16]. The properties of composition operator on the general Hardy spaces H 2 (γ) are studied in [7], [12], [15]. In this chapter, we investigate the properties of (α,β)- normal composition operators on general Hardy spaces H 2 (γ).

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For a sequence γ as above and a point w D, let Pin ∞ kw γ(z) = n=0 γ1n (wz )n 2

Then the function kw γ is a point evaluation for H 2 (γ) i.e., for f in H 2 (γ), (f, kw γ)γ = f (w) Then k0 γ =1 and Cφ∗ kw γ = kφ(w) γ. Theorem 3.1. If Cφ is (α,β)- normal on H 2 (γ) then α ≤

1 γ kkφ(0) k

≤ β.

Proof: Let Cφ be (α,β)- normal on H 2 (γ). By the definition of (α,β) normality, αkCφ∗ f kγ ≤ kCφ f kγ ≤ βkCφ∗ f kγ ∀f ∈ H 2 (γ) and if f = k0 γ, we have αkCφ∗ k0 γkγ ≤ kCφ k0 γkγ ≤ βkC ∗ k0 γkγ γ γ αkkφ(0) kγ ≤ kk0 γkγ ≤ βkkφ(0) kγ 1 α ≤ kkγ k ≤ β φ(0)

. Theorem 3.2. If Cφ is (α,β)- normal, then φ is univalent in the unit disk. Moreover, there is a subset E of the unit circle with measure zero, such that off E, the radial limits of φ exists and are distinct at distinct points. Proof: For non - constant φ, ker (Cφ ) = (0). If Cφ is (α,β)- normal, ker (Cφ ) = ker (Cφ∗ ), so = (0).

ker(Cφ∗ )

This implies ran (Cφ ) is dense. In particular, there is a sequence of polynomials pn so that Cφ pn = pn ◦ φ converges to z in H 2 (γ). Since pn ◦ φ(γ) converges to γ for each γ in the disk, φ is univalent in the disk. By possibly passing to a subsequence, we may assume that the boundary functions of the pn ◦ φ are defined and converge pointwise to z off a set E of measure zero. If eiθ1 and eiθ2 are distinct and not in E, then the convergence of the redial limit functions implies that infinitely many of the pn ◦ φ have distinct values at eiθ1 and eiθ2 which implies φ(eiθ1 ) and φ(eiθ2 ) are distinct.

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