Some Weak p-hyponormal Classes of Weighted Composition Operators

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1 Filomat :9 (07), Publishe by Faculty of Sciences an Mathematics, University of Niš, Serbia Available at: Some Weak p-hyponormal Classes of Weighte Composition Operators H. Emamalipour a, M. R. Jabbarzaeh a, M. Sohrabi Chegeni a a Faculty of Mathematical Sciences, University of Tabriz Abstract. In this note, we iscuss measure theoretic weighte composition operators on L (Σ) in some operator classes that are weaker than p-hyponormal.. Introuction an Preliminaries Let (X, Σ, µ) be a sigma finite measure space an let ϕ : X X be a measurable transformation such that µ ϕ is absolutely continuous with respect to µ. It is assume that the Raon-Nikoym erivative h = µ ϕ /µ is finite value or equivalently (X, ϕ (Σ), µ) is sigma finite. We use the notation L (ϕ (Σ)) for L (X, ϕ (Σ), µ ϕ (Σ)) an henceforth we write µ in place of µ ϕ (Σ). All comparisons between two functions or two sets are to be interprete as holing up to a µ-null set. We enote that the linear space of all complex-value Σ-measurable functions on X by L 0 (Σ). The support of f L 0 (Σ) is efine by σ( f ) = {x X : f (x) 0}. For a finite value function u L 0 (Σ), the weighte composition operator W on L (Σ) inuce by ϕ an u is given by W = M u C ϕ where M u is a multiplication operator an C ϕ is a composition operator on L (Σ) efine by M u f = u f an C ϕ f = f ϕ, respectively. Let A = ϕ (Σ). If ϕ (Σ) Σ, there exists an operator E := E ϕ (Σ) : L p (Σ) L p (A) which is calle conitional expectation operator. D(E), the omain of E, contains the set of all non-negative measurable functions an each f L p (Σ) with p, which satisfies f µ = E( f )µ, A ϕ (Σ). A A Recall that E : L (Σ) L (A) is a surjective, positive an contractive orthogonal projection. For more etails on the properties of E see [0, 4, 6]. Since by the change of variable formula, f ϕµ = h f µ, f L (Σ), X X then W f = he( u ) ϕ f. Put J = he( u ) ϕ. It follows that W is boune on L (Σ) if an only if J L (Σ) (see [] an also [4] for a iscussion of E( ) ϕ when ϕ is not invertible). Thus, W n 00 Mathematics Subject Classification. Primary 47B0 Seconary 47B8 Keywors. Conitional expectation, weighte composition operator, hyponormal, paranormal. Receive: 0 April 05 Accepte: 4 July 05 Communicate by Vlaimir Muller Corresponing author: M. R. Jabbarzaeh aresses: hemamali@tabrizu.ac.ir (H. Emamalipour), mjabbar@tabrizu.ac.ir (M. R. Jabbarzaeh), m.sohrabi@tabrizu.ac.ir (M. Sohrabi Chegeni)

2 H. Emamalipour et al. / Filomat :9 (07), is a boune operator precisely when J n := h n E n ( u n ) ϕ n L (Σ), where n N, h n = µ ϕ n /µ, u n = u(u ϕ)(u ϕ ) (u ϕ n ) an E n = E ϕ n (Σ). From now on, we assume that J L (Σ) an u 0. Put h 0 =, J = J, h = h an E = E. Let B(H) be the algebra of all boune linear operators on the infinite imensional complex Hilbert space H. Let T = U T be the polar ecomposition for T B(H), where U is a partial isometry an T = (T T) /. Definition.. Let n N, k N {0} an let p > 0. We enote by H(p, n, k) the set of all operators T on H that T k (T n T n ) p T k T k (T n T n ) p T k. Note that H(p, n, k ) H(p, n, k ), for all k k. Let r an q be two positive real numbers. An operator T is absolute (p, r)-paranormal if T p U T r x T r x p+r, an T is absolute (p, r)- -paranormal if T p U T r x r T r U x p+r, for all unit vectors x in H. An operator T is (p, r, q)-paranormal if T p U T r x q T p+r q x, an T is (p, r, q)- -paranormal if T p U T r x p+r q T q U x, for all unit vectors x H. Composition operators as an extension of shift operators are a goo tool for separating weak hyponormal classes. Classic seminormal (weighte) composition operators have been extensively stuie by Harrington an Whitley [9], Lambert [, 4], Singh [8], Campbell [4 6] an Stochel [8]. In [] an [] some weak hyponormal classes of composition operators are stuie. In those works, examples were given which show that composition operators can be use to separate each partial normality class from quasinormal through w-hyponormal. But in some cases composition operators can not be separate some of these classes. Hence, it is better that we consier the weighte case of composition operators. In [7] an [], the authors generalize the work one in [] an have obtaine some characterizations of relate p-hyponormal weighte composition operators as separately. In [] some examples were presente to illustrate that weighte composition operators lie between those classes. This note is a continuation of the work one in []. The plan of this note is to present some characterizations of weak p-hyponormal an weak p-paranormal classes of weighte composition operators on L (Σ). We then give specific examples illustrating these classes.. Characterizations In the following two theorems, to avoi teious calculations, we investigate only H(p,, k) an H(p, n, 0) classes of weighte composition operators. Note that T H(p,, 0) if an only if T is p-hyponormal an T H(p,, ) if an only if T is p-quasihyponormal. Recall that T H(p,, k) if an only if T k (T T) p T k T k (TT ) p T k, an T H(p, n, 0) if an only if (T n T n ) p (T n T n ) p. We assume throughout this note that W is a boune weighte composition operator on L (Σ). Theorem.. Let k N. Then W H(p,, k) if an only if Proof. Let f L (Σ). Then we have an E k (u k J p ) E k (u k u(h p ϕ)(e(u )) p E(uu k )), on σ(h k ). W k (W W) p W k f = h k E k (u k M J p(u k f ϕ k )) ϕ k f = h k E k (u k Jp ) ϕ k f, W k (WW ) p W k f = h k E k (u k u(h p ϕ)(e(u )) p E(uu k f ϕ k )) ϕ k f = h k E k (u k u(h p ϕ)(e(u )) p E(uu k )) ϕ k f.

3 Thus, W H(p,, k) if an only if H. Emamalipour et al. / Filomat :9 (07), h k E k (u k Jp ) h k E k (u k u(h p ϕ)(e(u )) p E(uu k )). Corollary.. Let C ϕ B(L (Σ)). Then C ϕ H(p,, k) if an only if E k (h p ) E k (h p ϕ) on σ(h k ). In particular (see []), if k =, then C ϕ is p-quasihyponormal if an only if E(h p ) h p ϕ on σ(h). Lemma.. [5] Let α an β be nonnegative an measurable functions. Then for every f L (Σ), if an only if σ(β) σ(α) an E n ( β α χ σ(α)). α f µ X Theorem.4. W H(p, n, 0) if an only if σ(u n ) σ(j n ) an Proof. Let f L (Σ). Then we have an E n (β f ) µ X (h p n ϕ n )(E n (u n )) p E n ( u n J p χ σ(jn )). n (W n W n ) p f, f = (W n W n ) p f, f = = X X J p n f µ, u n (h p n ϕ n )(E n (u n)) p (E n (u n f )) f µ E n ((h p n ϕ n )(E n (u n)) p X un f ) µ. Put α = J p n an β = (h p n ϕ n )(E n (u n)) p u n. Then σ(α) = σ(j n ) an σ(β) = σ(u n ). Now, the esire conclusion follows from Lemmamama.. Corollary.5. [7] W is p-hyponormal if an only if σ(u) σ(j) an (( ) p ) J ϕ u E. J E(u ) Lemma.6. Let T B(H) an let U T be its polar ecomposition. Suppose p, r an q are positive real numbers. Then the following hol: (i) [7] T is absolute (p, r)-paranormal if an only if for each λ > 0, r T r U T p U T r (p + r)λ p T r + pλ p+r 0. (ii) [] T is absolute (p, r)- -paranormal if an only if for each λ > 0, r T r U T p U T r (p + r)λ p T r + pλ p+r 0. (iii) [] T is (p, r, q)-paranormal if an only if for each λ > 0, T r U T p U T r qλ q T (p+r) q + (q )λ q 0. (iv) [] T is (p, r, q)- -paranormal if an only if for each λ > 0, T r U T p U T r qλ r(q ) T (p+r) q + (q )λ rq 0.

4 Theorem.7. The following statements are equivalent. (i) W is absolute (p, r)-paranormal. (ii) E(u J p ) (h p ϕ)(e(u )) p+, on σ(h) σ(e(u)). (iii) W is (p, r, q)-paranormal. Proof. (i) (ii) Let f L (Σ). It is easy to check that H. Emamalipour et al. / Filomat :9 (07), U f = u f ϕ h ϕe(u ) It follows that U f = h ((E(u )) E(u f )) ϕ W r f = J r f W r = u(h r ϕ)(e(u )) r E(u f ). W r U W p U W r f = rh r ((E(u )) r E(u J p )) ϕ f := a f. Now, by Lemmamama.6(i), W is absolute (p, r)-paranormal if an only if A(λ) := a (p + r)bλ p + pλ p+r 0, λ [0, ), where b = J r. Since this function takes its minimum value at λ = r b, then A(J) 0 a rj p+r rh r ((E(u )) r E(u J p )) ϕ r(h p+r (E(u )) p+r ϕ ) (h r ϕ)(e(u )) r E(u J p ) (h p+r ϕ)(e(u )) p+r, on σ(h) E(u J p ) (h p ϕ)(e(u )) p+, on σ(h) σ(e(u)). (iii) (ii) It is easy to check that W r U W p U W r f = h r ((E(u )) r E(u J p )) ϕ f W (p+r) q f = J p+r q f. By Lemmamama.6(iii), W is (p, r, q)-paranormal if an only B(λ) := a qλ q b + (q )λ q 0, λ [0, ), where a := h r ((E(u )) r E(u J p )) ϕ an b = J p+r q. This function has a minimum at λ = b. Then we have B(J p+r q ) 0 a b q h r ((E(u )) r E(u J p )) ϕ J p+r (h r ϕ)(e(u )) r E(u J p ) (h p+r ϕ)(e(u )) p+r, E(u J p ) (h p ϕ)(e(u )) p+, on σ(h) on σ(h) σ(e(u)). Hence the proof is complete.

5 H. Emamalipour et al. / Filomat :9 (07), Corollary.8. The following are equivalent. (i) C ϕ is absolute (p, r)-paranormal. (ii) E(h p ) h p ϕ on σ(h). (iii) C ϕ is (p, r, q)-paranormal. Note that by [, Theorem.], E(h p ) h p ϕ on σ(h) if an only if C ϕ is absolute p-paranormal if an only if C ϕ is p-paranormal. So for each r an q, these classes can not be separate by composition operators. Theorem.9. The following assertions hol. (i) W is absolute (p, r)- -paranormal if an only if on σ(h). (h r ϕ)(e(u )) r E(u J p ) (h p+r ϕ )({E(u) (p+r) r (E(u )) (p+r)(r ) r } ϕ) (ii) W is (p, r, q)- -paranormal if an only if on σ(h). (h r ϕ)(e(u )) r E(u J p ) (h p+r ϕ )({E(u) q (E(u )) (p+r q) } ϕ) Proof. (i) Let f L (Σ). Direct computations show that W r U W p U W r f = h r {(E(u )) r E(u J p )} ϕ f := C f, W r f = λ p u(h r ϕ)(e(u )) r E(u f ). Then by Lemmama ma.6 (ii), W is absolute (p, r)- -paranormal if an only if rc f (p + r)λ p u(h r ϕ)(e(u )) r E(u f ) + pλ p+r, f 0, for each λ (0, ). Put f = χ ϕ B with µ(ϕ B) <. Hence, (.) hols if an only if {rc (p + r)λ p u(h r ϕ)(e(u )) r E(u) + pλ p+r }µ 0. ϕ B Equivalently, {rc ϕ (p + r)λ p (E(u) ϕ )h r ((E(u )) r ϕ )(E(u) ϕ ) + pλ p+r }hµ 0. B But, this is equivalent to rc ϕ (p + r)λ p (E(u) ϕ ) h r (E(u )) r ϕ + pλ p+r 0 on σ(h). Set rc ϕ = a, an b = (E(u) ϕ ) h r (E(u )) r ϕ. Then W is absolute (p, r)- -paranormal if an only if D(λ) := a (p + r)bλ p + pλ p+r 0, λ [0, ).

6 Since it follows that on σ(h). D( r p+r b) 0 a rb r H. Emamalipour et al. / Filomat :9 (07), min λ [0, ) A(λ) = A( r b), (h r ϕ )((E(u )) r E(u J p )) ϕ (E(u) (p+r) r ϕ )h p+r ((E(u )) (p+r)(r ) r ϕ ) (h r ϕ)(e(u )) r E(u J p ) ({E(u) (p+r) r (E(u )) (p+r)(r ) r } ϕ)(h p+r ϕ ), (ii) The proof is similar to the proof of part (i). Let f L (Σ). By.6 (iv), W is (p, r, q)- -paranormal if an only if for each λ [0, ), where G(λ) := a qbλ r(q ) + (q )λ rq 0, a = (h r ϕ ){(E(u )) r E(u J p )} ϕ, b = (E(u) ϕ ) h p+r q ((E(u )) p+r q ϕ ). Since this function takes its minimum value at λ = r b, then we have G( r b) 0 a b q (h r ϕ )({(E(u )) r E(u J p )} ϕ ) ((E(u)) q ϕ )h p+r ((E(u )) (p+r q) ϕ ) (h r ϕ)(e(u )) r E(u J p ) ({(E(u)) q (E(u )) (p+r q) } ϕ)(h p+r ϕ ), on σ(h). This completes the proof. Corollary.0. The following are equivalent. (i) C ϕ is absolute (p, r)- -paranormal. (ii) (h r ϕ)e(h p ) h p+r ϕ on σ(h). (iii) C ϕ is (p, r, q)- -paranormal.. Examples Let {m n } n= be a sequence of positive real numbers. Consier the space l (m) = L (N, N, m), where N is the power set of natural numbers an m is a measure on N efine by m({n}) = m n. Let u = {u n } n= be a sequence of non-negative real numbers. Let ϕ : N N be a non-singular measurable transformation i.e. µ ϕ µ. Direct computation shows that (see [4]) h n (k) = m j m k j ϕ n (k) j ϕ E n ( f )(k) = n (ϕ n (k)) f j m j j ϕ n (ϕ n (k)) m j J n (k) = m k j ϕ n (k) for all non-negative sequence f = { f n } n= l (m) an k N. (u n (j)) m j,

7 H. Emamalipour et al. / Filomat :9 (07), Example.. Let X be the set of nonnegative integers, let Σ be the σ-algebra of all subsets of X, take m to be the point mass measure etermine by the m =,,, c,, c,, c,,, where c an are fixe positive real numbers. The powers of c occur for o integers an those of for even integers. Our point transformation ϕ is efine by 0 n = 0,, ϕ(n) = n n. Note that this example was use in [] an [] to show that composition operators can separate almost all weak hyponormality classes. Define u by n = 0 u(n) = c n (o) n (even). Then we get that Since then we have an so ( + c ) p n = 0 J p (n) = ( n+ ) p n (o) c n ( c n+4 ) p n (even), n (+c ) p +c p n = 0, E(u J p )(n) = c ( n+ ) p n (o) c n ( c n+4 ) p n (even), (E(u )) p+ (n) = (h ϕ)(n) = n ( +c c p+ p+ )p+ n = 0, n (o) n (even). j ϕ (ϕ(n)) m j m ϕ(n), p n = 0, (h p ϕ)(n) = ( c n ) p n (o) n (( c ) n ) p n (even), (+c ) p+ n = 0, (h p ϕ)(e(u )) p+ (n) = c p+ ( c n ) p n (o) n p+ (( c ) n ) p n (even). Then by Theorem.7, W is (p, r, q)-paranormal if an only if + c. Also, irect computations show that ( + c ) r {( + c ) p + c p } n = 0, (h r ϕ)(e(u )) r E(u J p )(n) = c r ( c n ) r ( c n+ ) p n (o) n n r (( c ) n ) r ( c n+4 ) p n (even), n

8 Set P(u) = ((E(u ) p+r q (E(u)) q ) ϕ. Then H. Emamalipour et al. / Filomat :9 (07), p+r n = 0,, (h p+r ϕ )(n) = ( c n ) p+r n (o) n 4 (( c ) n ) p+r n 4 (even), (P(u))(n) = (+c) q (+c ) p+r q (p+r) c (p+r) n = 0,, p+r+q n (o) n 4 (even), (+c) q (+c ) p+r q n = 0,, q {(h p+r ϕ )P(u)}(n) = (p+r) ( c n ) p+r n (o) n 4 c (p+r) (( c ) n ) p+r n 4 (even). Then by Theorem.9, W is (p, r, q)- -paranoramal if an only if Now, + c ( + c )p ( + c ( + c) ){ ( + c ) }q. () n = 0 u (n) = u(n)u(ϕ(n)) = c n = (c) n. Take Q(u) = (uu )E(uu )(h p ϕ)(e(u )) p. Then n = 0 c n = u(n)u (n) = (c ) n (o) (c ) n (even), +c n = 0, (c ) n (o) E(uu )(n) = (c ) n (even). ( + c ) p n = 0 c ( + c ) p n = (Q(u))(n) = c p+ ( c n ) p n (o) n p+ c (( c ) n ) p n (even), an so Also, it is easy to check that (+c ) p +c (+c ) p +c p+ n = 0,, E (Q(u))(n) = c p+ ( c n ) p n (o) n p+ c (( c ) n ) p n (even).

9 H. Emamalipour et al. / Filomat :9 (07), Then by Theorem., W H(p,, ) if an only if ( + c ) p n = 0 c p n = u Jp (n) = c ( n+ ) p n (o) c n c ( c n+4 ) p n (even). n (+c ) p +c p +(c c p ) n = 0,, E (u Jp )(n) = c ( n+ ) p n (o) c n c ( c n+4 ) p n 4 (even). n + ( + c ) p ( c )p. () By similar computations we have ϕ 0 n = 0,, (n) = n n, n = 0,, h (ϕ (n)) = c n (o) n 4 (even), +c +(c) E (u n = 0,, )(n) = (c) n (o) (c) n 4 (even), + c + (c) n = 0 J (n) = (c) c n (o) (c) n (even), u J p E ( u J p )(n) = n = 0 (+c +(c) ) p c n = (c) (n) = p c p (c) n (o) (c) p c p (c) n (even), (c) p p (+c +(c) ) p + c (c) p c p + (c) (c) p p (c) (c) p c p (c) (c) p p n = 0,, n (o) n 4 (even). Put R(u) = (h p ϕ )(E (u ))p E ( u ). Then J p ( + c + (c) ) p { + c + (c) } n = 0,, (R(u))(n) = (+c +(c) ) p c p (c) p p (c) p n.

10 Then by Theorem.4, W H(p,, 0) if an only if H. Emamalipour et al. / Filomat :9 (07), ( + c + (c) ) p { ( + c + (c) ) + c p c p (c) + (c) }. () p p (c) p In particular, if p = q = an c = (0, 0.5), then by (), (), (), W is (p, r, q)- -paranormal but W neither in H(p,, ) nor in H(p,, 0). Also if c = (0.5,.), then W is not (p, r, q)- -paranormal but W is in H(p,, ) H(p,, 0). Now let n = 0, u(n) = n n. Then we have n = 0, u (n) = n = n(n ) n, E (u )(n) = n = 0,, (n(n )) n, ( u (E (u ))p J p )(n) = E ( u (E (u ))p J p )(n) = 6 n = 0 J (n) = c((n + )(n + )) n (o) ((n + )(n + )) n (even), n = 0 p p p+ c n = p p p n = p n(n ) ( (n+)(n+) )p c n (o) p n(n ) ( (n+)(n+) )p n 4 (even), p ( ( ( p )+( p p+ c p )+( p p p ) n(n ) (n+)(n+) )p c p n(n ) (n+)(n+) )p p n = 0,, n (o) n 4 (even). Put S(u) = (h ϕ ) p (E (u ))p E ( u χ σ(j ) ). Then J p 6 (S(u))(n) = + + n = 0,, p+ p+ c p p p+ p n(n ) ( (n+)(n+) )p n. Then by Theorem.4, W H(p,, 0) if an only if ( 4 6c )p + 4( 6 )p 5 4 p. (4) Put T(u) = (u u)(h p ϕ)(e(u )) p E(u u). Since p n = 0, (h p ϕ)(n) = {( c ) n } p n (even) { c n } p n (o), n

11 H. Emamalipour et al. / Filomat :9 (07), n = 0 J(n) = (n + ) c n n (even) n (n + ) ( c ) n n (o), p n = 0 p n = u Jp (n) = p c p n = (n(n )) (n + ) p ( c n ) p n 4 (even) n (n(n )) (n + ) p (( c ) n ) p n (o), Hence p + p +( p c p ) n = 0,, E (u Jp )(n) = n (n ) (n + ) p ( c n ) p n 4 (even) n n (n ) (n + ) p (( c ) n ) p n (o), p n = 0, (T(u))(n) = n p+ (n ) (( c ) n ) p n (even) n p+ (n ) ( c n ) p n (o), n an so p + p + p+ n = 0,, E (T(u))(n) = n p+ (n ) (( c ) n ) p n 4 (even) n p+ (n ) ( c n ) p n (o). n Then by Theorem., W H(p,, ) if an only if 4( 9 4 c)p ( )p, (5) an by Theorem.7, for each p an r, W is absolute (p, r)-paranormal. Again, by similar computations we have E(u n = 0, )(n) = n n, p+r n = 0, (h p+r ϕ)(n) = (( c ) n ) p+r n (even) ( c n ) p+r n (o), n p n = 0 p n = (u J P )(n) = n (n + ) p ( c n ) p n (even) n n (n + ) p (( c ) n ) p n (o),

12 Then H. Emamalipour et al. / Filomat :9 (07), p + p n = 0, E(u J P )(n) = n (n + ) p ( c n ) p n (even) n n (n + ) p (( c ) n ) p n (o). r ( p + p ) n = 0, (h r ϕ)(e(u )) r E(u J P )(n) = n r (n + ) p ( c n ) p (( n c ) n ) r n (even) n r (n + ) p (( c ) n ) p ( c n ) r n (o). n p+r n = 0,, (U(u))(n) = (n ) (p+r) (( c ) n ) p+r n 4 (even) (n ) (p+r) ( c n ) p+r n (o), where U(u) = (h p+r ϕ )((E(u)) (p+r) r (E(u )) (p+r)(r ) r an only if n 4 ϕ). Thus by Theorem.9, W is absolute (p, r)-( )-paranormal if r ( 9c )p. (6) Accorollarying to (4), (5) an (6), if p = q = r = an c = (0, 0.), then W H(p,, 0) \ H(p,, ), W is absolute (p, r)-paranormal but it is not absolute (p, r)- -paranormal operator. However if c = (5, ), then W is in H(p,, ) \ H(p,, 0) an W not only is absolute (p, r)-paranormal but it is also absolute (p, r)- -paranormal. Example.. Let X = [0, ] equippe with the Lebesgue measure µ on the Lebesgue measurable subsets an let ϕ : X X is efine by x 0 x ϕ(x) =, x x. Then an so h(x) = h (x) = an for each f L (Σ) Put f (x) + f ( x) E( f )(x) =, 4x 0 x 4 ϕ 4x 4 (x) = x + 4x x 4 4 4x 4 x, (E( f ) ϕ )(x) = f ( x ) + f ( x ), f (x)+ f ( x)+ f ( x)+ f ( +x) 4 0 x E ( f )(x) =, f (x)+ f ( x)+ f ( +x)+ f ( x) 4 x <. 0 0 x 4 u(x) = x 4 x, an take V(u) = E (u u(h p T)(E(u )) p E(u u)). Then 0 0 x 4 u (x) = x( x) 4 x,

13 Hence we obtain H. Emamalipour et al. / Filomat :9 (07), x 4 J (x) = 4x ( x) 4 x. E(u 0 0 x 4 )(x) = x x+ 4 x, 0 0 x 4 E(u u)(x) = x( x) 4 x. 0 0 x E (u Jp 4 )(x) = x ( x) {(x x+) p +(+x ) p }+( +x) ( x) {(x x+ 4 )p +(x x+ 5 4 )p } p 4 x, 0 0 x 4 V(u)(x) = x ( x) (x x+) p +( +x) ( x) (x 4x+ 5 )p p 4 x. Thus by Theorem., W H(p,, ) if an only if E (u Jp ) V(u), i.e., x ( x) {(x x + ) p + ( + x ) p } + ( + x) ( x) {(x x + 4 )p + (x x )p } p {x ( x) (x x + ) p + ( + x) ( x) (x 4x + 5 )p }. But the above inequality is not hols on X = [0, ]. So W is not in H(p,, ). Also, by Theorem.7, W is (p, r, q)-paranormal if an only if By Theorem.9, W is (p, r, q)- -paranormal if an only if x (x x + ) p + ( x) ( + x ) p p (x x + ) p+. (7) p (8x x + 5) p+r q q (x x + ) r {x (x x + ) p + ( x) ( + x ) p }. (8) In particular, if p = q = 00 an r =, then by (7) an (8), W is (p, r, q)- -paranormal but W is not (p, r, q)- paranormal. However if p = 0.5, q =, r =, then W is (p, r, q)-paranormal but W is not (p, r, q)- -paranormal. Acknowlegment The authors woul like to thank the referee for very helpful comments an valuable suggestions. References [] N. L. Braha an I. Hoxha, (p, r, q)- -Paranormal an absolute-(p, r)- -paranormal operators, J. Math. Anal. 4 (0), 4-. [] C. Burnap an I. Jung, Composition operators with weak hyponormalities, J. Math. Anal. Appl. 7 (008), [] C. Burnap, I. Jung an A. Lambert, Separating partial normality classes with composition operators, J. Operator Theory 5 (005), [4] J. T. Campbell an J. Jamison, On some classes of weighte composition operators, Glasgow Math. J. (990), [5] J. T. Campbell, M. Embry-Warrop an R. Fleming, Normal an quasinormal weighte composition operators, Glasgow Math. J. (99), [6] J. Campbell an W. Hornor, Localising an seminormal composition operators on L, Proc. Roy. Soc. Einburgh Sect. A 4 (994), 0-6. [7] M. Cho an T. Yamazaki, Characterizations of p-hyponormal an weak hyponormal weighte composition operators. Acta Sci. Math. (Szege) 76 (00), 7-8.

14 H. Emamalipour et al. / Filomat :9 (07), [8] A. Daniluk, an J. Stochel, Seminormal composition operators inuce by affine transformations, Hokkaio Math. J. 6 (997), [9] D. Harrington an R. Whitley, Seminormal composition operators, J. Operator Theory (984), 5-5. [0] J. Herron, Weighte conitional expectation operators, Oper. Matrices 5 (0), [] T. Hoover, A. Lambert an J. Queen, The Markov process etermine by a weighte composition operator, Stuia Math. (Polan) LXXII (98), 5-5. [] M. R. Jabbarzaeh an M. R. Azimi, Some weak hyponormal classes of weighte composition operators, Bull. Korean. Math. Soc. 47 (00), [] D. Jung, M. Y. Lee an S. H. Lee, On classes of operators relate to paranormal operators, Sci. Math. Jpn. 5 (00), -4. [4] Alan Lambert, Localising sets for sigma-algebras an relate point transformations, Proc. Roy. Soc. Einburgh Sect. A 8 (99), -8. [5] A. Lambert, Hyponormal composition operators, Bull. Lonon Math. Soc. 8 (986), [6] M. M. Rao, Conitional measure an applications, Marcel Dekker, New York, 99. [7] D. Senthilkumar an P. Maheswari Naik, Weyl s theorem for algebraically absolute-(p, r)-paranormal operators, Banach J. Math. Anal. 5 (0), 9-7. [8] R. K. Singh an J. S. Manhas, Composition operators on function spaces, North Hollan Math. Stuies 79, Amsteram 99.