Kuldip Raj and Sunil K. Sharma. WEIGHTED SUBSTITUTION OPERATORS BETWEEN L p -SPACES OF VECTOR-VALUED FUNCTIONS. 1. Introduction and preliminaries

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1 F A S C I C U L I M A T H E M A T I C I Nr 47 0 Kuldip Raj and Sunil K. Sharma WEIGHTED SUBSTITUTION OPERATORS BETWEEN L p -SPACES OF VECTOR-VALUED FUNCTIONS Abstract. In this paper we characterize weighted substitution operators between L p -spaces of vector-valued functions and also make an attempt to characterize isometry and partial isometry of these operators. Key words: weighted substitution operator, isometry, partial isometry, adjoint of an operator. AMS Mathematics Subject Classification: Primary 47B0, Secondary 47B38.. Introduction and preliminaries Let (, S, µ) be a σ-finite measure space. Then for p <, L p (, C n ) denotes the class of all S-measurable C n -valued functions whose pth power is integrable on with respect to the measure µ i.e. L p (, C n ) { f f : C n Then L p (, C n ) is a Banach space under the norm, ( f } is a measurable and f( ) p dµ <. ) f( ) p p dµ and for p, L (, C n ) is a Hilbert space under the inner product, f, g f( ), g( ) dµ. Let w : C n be a vector-valued measurable function and let T : be a non-singular measurable transformation. Then a bounded linear transformation S w,t : L p (, C n ) L p (, C n ) defined by (S w,t, f)(x) w(x)f(t (x))

2 64 Kuldip Raj and Sunil K. Sharma is called a weighted composition operator or a weighted substitution operator induced by the pair (w, T ). If we take w(x), the constant one function on, we write S w,t as C T and call it a composition operator or substitution operator induced by T. In case T (x) x, for every x, we write S w,t as M w and call it a multiplication operator induced by w. An atom of a measure µ is an element A S, if F A then either µ(f ) 0 or µ(f ) µ(a). A measure with no atoms is called non atomic. We can easily check the following well known facts see [5]. (a) Every σ-finite measure space (, S, µ) can be decomposed into disjoint sets B and Z, such that µ is non atomic over B and Z is atmost countable union of atoms A n of finite measure. So we can write as follows: B ( n N {A n }). (b) For each f L s (, S, µ), there exists two functions f L p (, S, µ) and f L q (, S, µ) such that f f f and f s s f p p f q q where p + q s. (c) Suppose p < q <. If a S 0 -measurable set K, is non-atomic and s.t. µ(k) > 0, there exists a function g 0 L p (, S 0, µ) with K g 0 q dµ. Let (, S, µ) be a σ-finite measure space and S 0 S be a σ-finite subalgebra. Then the conditional expectation E( S 0 ) is defined as a linear transformation from certain S-measurable function spaces (i.e. L, L etc) into their S 0 -measurable counterparts. In particular the conditional expectation with respect to the σ-algebra T (S) is a bounded projection from L p (, S, µ) onto L p (, T (S), µ). We denote this transformation by E. The transformation E has the following properties: (i) E(f g T ) E(f) (g T ) (ii) if f g almost everywhere, then E(f) E(g) almost everywhere (iii) E() (iv) E(f) has the form E(f) g T for exactly one σ-measurable function g. In particular g E(f) T is a well defined measurable function. (v) E(fg) (E f )(E g ) (vi) For f > 0 almost everywhere, E(f) > 0 almost everywhere. (vii) If φ is a convex function, then φ(e(f)) E(φ(f)) µ-almost everywhere. For deeper study of the properties of E see []. Campbell ([], []) made use of the expectation operator to study some properties of weighted composition operators on L (, C). Also T is a mapping from into itself is a non-singular measurable transformation such that µ T is absolutely continuous with respect to µ (i.e. µ T µ). Hence by Radon-Nikodym derivative theorem there exists a positive measurable function f 0 such that µ(t (E)) f 0 dµ, for every E S. The function E

3 Weighted substitution operators between f 0 is called the Radon-Nikodym derivative of the measure µt with respect dµt to the measure µ. It is denoted by dµ. Boundedness of the composition operators in L p (, S, µ), ( p < ) spaces, where the measure spaces are σ- finite, appeared already in [3] and for two different L p -spaces in [4]. Also boundedness of weighted operators on C(, E) has already been studied in [9]. More detailed classes of weighted composition operators on some function spaces are considered in ([3], [4], [5], [6], [9], [0], []). In this paper we plan to study weighted composition operators on vector valued L p -spaces.. Weighted substitution operators Theorem. Suppose p, q <. Every weighted substitution transformation S w,t : L p (, C n ) L q (, C n ) is always bounded. Proof. It is easy to prove by using closed graph theorem and so we omit it. Theorem. Let S w,t : L (, C n ) L (, C n ) be a linear transformation. Then S w,t is bounded if and only if J L (, C n ), where J f 0 E n n (w w) T. Proof. The proof is given by Hornor and Jamison [[6], p-34]. In the next theorem, we characterize the boundedness of weighted substitution operator for atomic measure spaces. Theorem 3. S w,t C(L (N, C n )) if and only if J : N C n n is a bounded function, where J(n) m T ({n}) Proof. For any f L (N, C n ), consider µ(m)w (m)w(m). µ(n) S w,t f (w f T )(n), (w f T )(n) µ(n) n w(n)f(t (n)), w(n)f(t (n)) µ(n) n n m T ({n}) w(m)f(n), w(m)f(n) µ(m)

4 66 Kuldip Raj and Sunil K. Sharma n n m T ({n}) m T ({n}) J(n)f(n), f(n) µ(n) n M J f. w (m)w(m)f(n), f(n) µ(m) µ(m)w (m)w(m) f(n), f(n) µ(n) µ(n) Hence S w,t is a bounded operator if and only if J : N C n n is a bounded function. Theorem 4. Let S w,t C(L (N, C n )). Define (Ag)(n) µ(n) Then S w,t A. m T ({n}) Proof. For any f, g L (N, C n ), consider S w,t f, g µ(m)w (m)g(m) for every g L (N, C n ). (w f T )(n), g(n) µ(n) n n n m T ({n}) m T ({n}) f(n), n w(m)f(t (m)), g(m)) µ(m) f(n), w (m)g(m)) µ(m) m T ({n}) f(n), (Sw,T g)(n) µ(n) n f, Ag. µ(m)w (m)g(m) µ(n) µ(n) Hence S w,t A. Theorem 5. Let S w,t C(L (, C n )). Then S w,t is a partial isometry if and only if J is an idempotent.

5 Weighted substitution operators between Proof. Suppose S w,t is a partial isometry. Then and therefore or S w,t S w,t S w,t S w,t S w,t S w,t S w,t S w,t S w,t S w,t M J M J. Hence we can conclude that J is an idempotent. Conversely, if J is an idempotent mapping, then,since kers w,t kerm J, so for any f (kers w,t ) ranm J, we have Sw,T S w,t f, g w f T, w g T dµ w w f T, g T dµ he n (w w) T f, g dµ Jf, g dµ f, g. Hence S w,t is a partial isometry. Theorem 6. Let S w,t : L (, C n ) L (, C n ) be a bounded operator. Then S w,t is an isometry if and only if J (x) is an isometry for µ-almost all x. Proof. The proof follows from the equality, S w,t f M J f for every f (, C n ). Theorem 7. Let S w,t C(L p (, C n )). Then S w,t is an idempotent operator if and only if w w T w and T T on supp w supp (w T ). Proof. Suppose S w,t is an idempotent operator. Then for e k C n, we have for any E S, with µ(e) <, which implies that S w,t S w,t (χ E e k ) S w,t (χ E e k ), w w T (χ (T ) (E)e k ) w(χ T (E)e k ).

6 68 Kuldip Raj and Sunil K. Sharma Hence T T and w w T w on supp w supp w T. The converse is easy to prove. Example. Let R, T (x) x + and w(x) { e (x ), for x < 0, for x. Then T (S) S, f 0, E[f] f for every f L p (, C n ). Now q f0 w T r r w(x ) r dµ. Hence S w,t is a weighted substitution operator from L p (, C n ) into L p (, C n ) in view of Theorem. Example. Let [0, ],... T (x) { x, if 0 x x, if < x. And w(x) x for every x. Then f 0 almost everywhere, so (EF )(x) [f(x) + f( x)] Now E(f) T (x) [ f(x) + f ( x )]. q f0 E( w T ) r r 0 0 E( w T (x) r dµ w( x x ) + w( ) r dµ. Hence S w,t is a bounded operator. If w(x) is an isometry for almost all x, then we present a characterization for boundedness of weighted substitution operators by using the Radyon - Nikodym derivative. Theorem 8. Let w : C n n be a measurable function and let T : be a non-singular transformation. Then for p, q <, S w,t : L p (, C n ) L q (, C n ) is continuous if and only if v L (, C n ).

7 Weighted substitution operators between Proof. For each E S, set λ(e) w(x) q dµ(x). Then T (E) S w,t f q w(x)(f T )(x) q dµ(x) w(x) q f T (x) q dµ(x) f(t (x)) q dλ(x) f(x) q dλt f(x) q v(x)dµ(x), where v dλt dµ. Hence, we can conclude that S w,t is a bounded linear transformation if and only if v is essentially bounded. References [] Campbell J.T., Jamison J.E., On some classes of weighted composition operators, Glasgow Math. J., 3(990), [] Campbell J.T., Jamison J.E., The analysis of composition operators on L p and Hopf Decomposition, J. Math. Anal. Appl., 59(99), [3] Carlson J.W., Weighted Composition Operators on l, Ph.D. Thesis, Purdue Univ., 985. [4] Carlson J.W., Hyponormal and quasinormal weighted composition operators, Rocky Mountain Journal of Mathematics, 0(990), [5] Ding J., Horvor W.E., A new approach to Frobenius-Perron operators, J. Math. Analysis Applicable, 87(994), [6] Hornor W.E., Jamison J.E., Weighted compositio operator on Hilbert spaces of vector valued functions, Proc. Amer. Math. Soc., 4(996), [7] Hoover T., Lambert T., Quim J., The Morkov Process determined by a weighted compostion operator, Studia Math., LII(98), [8] Jabbarzadeh M.R., Pourreza E., A note on weighted composition operator on L p spaces, Bull. Iranian Mathematical Society, 9()(003), [9] Jamison J.E., Rajagopalan M., Weighted compostion operators on C(, E), Journal of Operator Theory, 9(988), [0] Kamowitz H., Compact weighted endomorphism of C(), Proc. Amer. Math. Soc., 83(98), [] Komal B.S, Raj K., Gupta S., On operators of weighted substitution on the generalized spaces of entire functions, I. Math. Today, 5(997), 3-0. [] Lambart A., Localising sets for sigma-algebras and related point transformations, Proc. Royal Soc. of Eolinburgh, ser. A, 8(9), -8.

8 70 Kuldip Raj and Sunil K. Sharma [3] Singh R.K., Composition operators induced by rational functions, Proc. Amer. Math. Soc., 59(976), [4] Takagi H., Yokouchi K., Multiplication and composition operators between two L p -spaces, Contem. Math., 3(999), [5] Zaanen A.C., Integration, nd ed., North Holland Amsterdam, 967. Kuldip Raj School of Mathematics, Shri Mata Vaishno Devi University Katra - 830, J&K, India kuldeepraj68@rediffmail.com Sunil K. Sharma School of Mathematics, Shri Mata Vaishno Devi University Katra - 830, J&K, India sunilksharma4@yahoo.co.in Received on and, in revised form, on