Spectral Properties of p – hyponormal Composition Operators

Far East J. Math. Sci. (FJMS) 9(3) (2003), 287 – 292. SPECTRAL PROPERTIES OF p-HYPONORMAL COMPOSITION OPERATORS S. PANAYAPPAN and D. SENTHIL KUMAR (Received December 3, 2002) Submitted by Alan Lambert Abstract In this paper, spectral properties of p –hyponormal composition operators acting on an L2- space are studied. 1. Definitions and Preliminaries Let ( X, Σ, λ ) be a sigma – finite measure space. Let T : X X be a - 1 measurable transformation with the property that T is absolutely continuous with respect to . We denote the Radon - Nikodym derivatives of T- 1 with respect to by h. Further let h be the finite valued a.e. If W : X R is a non – negative measurable function, then the weighted composition operator WT induced by T and w is given by WTf(x) = w(x) . f T(x), f ∈ L2( X, Σ, λ ). [7, 8] In case that w = 1 a.e., we say that WT is a composition operator denoted by CT. 2000 Mathematics Subject Classification: 47B20, 47B38. Key words and phrases: Hilbert spaces, p – hyponormal composition operators. © 2003 Pushpa Publishing House 288 S. PANAYAPPAN and D. SENTHIL KUMAR Let B(H) denote the algebra of all bounded linear operators on a Hilbert space H. An operator T ∈ B(H) is sid to be p – hyponormal if (T* T)p (T T*)p. If p = 1, T is hyponormal and if p = ½, T is semi – hyponormal. It is well known that a p – hyponormal operator is q – hyponormal for q p by Lowner’s theorem. The p –hyponormal operators were first studied by Aluthge [1]. For an operator T, we denote the spectrum, point spectrum and approximate point spectrum by (T), p(T) and a(T), respectively. A point Z ∈ C is in the joint point spectrum jp(T) if there exists a non – zero vector x such that Tx = Zx and T*x = Zx. The set of all complex numbers is denoted by C and the unit circle by c. 2. Results THEOREM 1. Let CT be p-hyponormal. Then 2 σ ( CT ) ⊂ { z ∈ C /  z  ∈ E R ( h ) }, where ER ( h ) is the essential range of h. Proof. We have CT* CT = Mh, the multiplication operator induced by h. So, σ (CT* CT ) = ER ( h ). Since CT is p-hyponormal , if r ∈ ER ( h ), then there exists z ∈ σ ( CT ) such that  z 2 = r [ 3 ]. Hence, σ ( CT ) ⊂ {z ∈ C /  z 2 ∈ ER ( h ) }. It must be noted that for non p-hyponormal CT, the result need not be true. For example, consider a unilateral shift for CT. Its spectrum is not contained in C ∪ { 0 }. Further, the inclusion relation of the theorem need not be an equality for p-hyponormal CT as shown in Corollary 3. … p –HYPONORMAL COMPOSITION OPERTAORS 289 Corollary 1. Let CT be a hyponormal. Then σ ( CT ) ⊂ { z ∈ C / 2 z  ∈ ER ( h ) }. Proof. CT is semi-hyponormal iff CT is hyponormal [ 7 ]. Corollary 2. Let WT be a p-hyponormal. Then σ ( WT ) ⊂ { z ∈ C /  z 2 ∈ ER ( J ) }. Proof. For a weighted composition operator WT, we have WT* WT = MJ. Corollary 3. Let CT be p-hyponormal with σ ( CT ) ⊂ R. Then σ (CT) = {1,-1}. Proof. It is well known that a p-hyponormal operator with a real spectrum is self adjoint [ 9 ]. Thus, h = 1 a.e. So, σ ( CT ) ⊂ {1, -1 }. But, σ ( CT ) = { 1 } means CT is an identity and σ ( CT ) = { -1 } is an impossibility. Theorem 2. Let CT be a p - hyponormal composition operator. Then 2 σp ( CT ) ⊂ { z ∈ C /  z  = h }. Proof. z ∈ σjp ( CT ) if and only if there exists a non-zero f such that T f = z f _ z ∈ σjp ( C T) iff there exists a non-zero f and T * f = z f. Equivalently, 2 such that T * T f =  z  f. By [ 9 ], if T be a p-hyponormal for 0 < p < ½, σjp ( T ) = σp ( T ). Since CT* CT = Mh, we have CT* CT f = h f and so σp ( CT ) 2 ⊂ { z ∈ C /  z  = h}. Ridge [ 10 ] has shown that σa ( CT ) - c and σp ( CT ) - c have circular symmetry about 0. This fact is used to prove the following result which relates the spectra of C * and C . T T 290 S. PANAYAPPAN and D. SENTHIL KUMAR Theorem 3. Let CT be p - hyponormal and 1 ∉ ER ( h ). Then (i) σp ( CT ) ⊂ σp ( CT* ) , and (ii) σ ( CT ) ⊂ σ ( CT* ). Proof. Let 1 ∉ ΕR ( h ). (i) z ∈ σ p ( CT ) z ∈ σ p ( CT ) CT f = z f CT* f = z f [2 ] z ∈ σp ( CT* ). (ii) Since CT is a p-hyponrmal, σ ( CT ) = σa ( CT* )*. As 1 ∉ ΕR ( h ), σ ( CT )* = σ ( CT ) and so σ ( CT ) = σa ( CT* ) ⊂ σ ( CT* ). The next theorem shows that the Weyl spectrum W(CT) and the spectrum σ ( CT ) can differ only on the unit circle when CT is the p-hyponormal. Theorem 4. Let CT be p-hyponormal with 1 ∉ ER ( h ). Then σ ( CT ) = W ( CT ). Proof. For p-hyponormal operators Weyl’s theorem holds [ 5]. So, W ( CT ) = σ ( CT ) - π00 ( CT ), where π00 ( CT ) is the set of isolated eigenvalues of finite multiplicity of σ ( CT ). First, we claim that 0 ∉π00(CT). Note that Weyl’s theorem holds for T, if and only if zo ∈ π00 ( T ) implies that ran ( T - zo ) is closed [ 6 ]. So, if 0 ∈ π00 ( CT ), then CT has closed range and so CT is invertible, a contradiction. Let λ ∈ π00 ( CT ), where λ ≠ 0. Then by [ 5 ], λ is an eigen value of CT. Since σp ( CT ) – c has circular symmetry about 0, λ cannot be isolated. Hence π00 ( CT ) = . In [ 10 ], it is shown that σp ( CT ) – { 0 } is a finite subset of c, when CT is compact. The next theorem sharpens the results. … p –HYPONORMAL COMPOSITION OPERTAORS 291 Theorem 5. Let CT be compact. Then σ ( CT ) = σp ( CT ). Proof. If T is compact, it is known that σ ( T ) is atmost countable and has zero as an accumulation point (possible) and if s ∈ σ ( T ), s ≠ 0, then s is an eigen value [ 6 ]. Since CT is compact, σ ( CT ) ⊂ { 0 } ∪ c and 0 cannot be an accumulation point of σ ( CT ). Thus, σ ( CT ) ⊂ c and all its elements are eigen values. When µ is non-atomic it is known that there is no compact composition 2 operator on L ( µ ) [11]. The following result is on infinite 2 dimensional L ( µ ). 2 Corollary 4. Let L ( µ ) be infinite dimensional. Then no 2 p-hyponormal composition operator on L ( µ ) is compact. Proof. Suppose CT is p-hyponormal and compact. Then it must be normal [ 3 ]. Also by Theorem 5, σ ( CT ) ⊂ c. But, a normal operator is unitary iff the spectrum is contained in the unit circle and so CT is unitary, a contradiction. Acknowledgement The first author acknowledges the receipt of grant no. F4-5/96 (Minor/SRO) from the University Grants Commission, Hyderabad, in support of this work. References [1] A. Aluthge, p – hyponormal operators for 0 < p < 1, Integral Eqns. Oper. Theo. 13 (1990), 307 – 315. [2] M. Cho and T. Huruya, p – hyponormal operators for 0 < p < ½, Comment. Math. 33 (1993), 23 – 29. [3] M. Cho amd masuo itoh, Putnam’s inequality for p - hyponormal operators, Proc. Amer. Math. Soc. 123 (1995), 2435 – 2440. 292 S. PANAYAPPAN and D. SENTHIL KUMAR [4] M. Cho and Hirohiko Jin, On p – hyponormal operators, Nihonkai Math. J. 6 (1995), 201 – 206. [5] M. Cho, masuo Itoh and Sartoru Oshio, weyl’s theorem holds for p – hyponormal operators, Glasgow Math. J. 39 (1997), 217 – 220. [6] Vasile Istratesscu, Introduction to Linear operator Theory, Marcel Dekker. Inc., New York, Basel, 1981. [7] T. James Campbell and E. Willam Horner, Localising and seminormal composition operators on L2, Proc. Roy. Soc. Edinburgh 124 A (1994), 301 – 316. [8] S. panayappan, Non – hyponormal weighted composition operators, Indian. J. Pure Appl. Math. 27 (10) (1996), 979 – 983. [9] S. M. Patel, A note on p- hyponormal operators for 0 < p < 1, Integral Eqns. Oper. Theo. 21 (1995), 498 – 503. [10] W. C. Ridge, Spectrum of a composition operator, Proc. Amer. Math. Soc. 37 (1973), 121 - 129. [11] R. K. Singh and Ashok kumar, Compact composition operators, J. Austral. Math. Soc. (Series A) 28 (1979), 309 – 314. Department of Mathematics Government Arts College Coimbatore – 641 018 Tamil Nadu, India Department of Mathematics Sri Ramakrishna Engineering College Coimbatore – 641 022 Tamil Nadu, India