Volterra Composition Operators

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 6, 211, no. 7, Volterra Composition Operators Anupama Gupta G.C.W. Parade, Jammu, India anu B. S. Komal Department of Mathematics University of Jammu, Jammu, India Abstract In this paper we characterized the Volterra composition operator. It is shown that the spectrum of Volterra composition operator is consisting of zero only. Mathematics subject classification: Primary 47B2, Secondary: 46B38 Keywords: Composition operator, Volterra operator, Expectation operator, Idempotent oprator, Radon-Nikodym derivative 1 Preliminaries Let (X, S, μ) be a s-finite measure space and let φ : X X be a non-singular measurable transformation (μ(e) μφ 1 (E) ). Then a composition transformation, C φ : L p (μ) L p (μ), 1 p<, is defined by C φ f foφ for every f L p (μ). In case C φ is continuous, we call it a composition operator induced by φ. It is easy to see that C φ is a bounded operator if and only if dμφ 1 f dμ o, the Radon-Nikodym derivative of the measure μφ 1 with respect to the measure μ, is essentially bounded. For more detail about composition operators we refer to Nordgen [4],Shapiro[7],Cowen[3],Singhand Manhas[11],Singh and Komal [1]. For each f L p (μ), 1 p<, there exists a unique φ 1 (S) measurable function E(f) such that gfdμ ge(f)dμ for every φ 1 (S) measurable function g for which left integral exists. The function E(f) is called conditional expectation of f with respect to the sub algebra φ 1 (S). In particular E(f) goφ if and only if g (E(f)oφ 1 ). For more

2 346 A. Gupta and B. S. Komal properties of the expectation operator, see Parthasarthy [8 ]and Lambert [2]. The Volterra operator V is an integral operator induced by the kernel K(x,y) defined as {, if x y K(x, y) 1, if x>y i.e. (Vf)(x) f(y)dy y Let φ :[, 1] [, 1] be a measurable transformation. The Volterra composition operator V is defined by (V φ f)(x) (C φ Vf)(x) φx f(t)dtfor every f L p [, 1] Set K φ (x, y) f o (x)e(φ 1 (y)) 2 dμ(y)dμ(x). A great deal of work on voltera operators was done by mathematicians Sinnamen [5], Erdos [6],Stepanov ([13],[14]) Sunder and Halmos [9], Lybic s [15] conjecture was introduced by Whitley [12] and generalized it to Volterra composition operators on L p [, 1].Gupta and Komal [3] also studied composite integral operator on L p spaces. Our main purpose in this paper is to study Volterra composition operator. It is shown that spectrum of Volterra composition operator is equal to {}. Theorem 1.1 Suppose K φ L 2 (μ μ). Then V φ is bounded Volterra composition operator. Proof: For any f L 2 (μ), we have V φ f 2 (V φ f)(x) 2 dμ(x) f(φ(x))dμ(y) 2 dμ(x) ( f o (x)e(φ 1 (y)) 2 dμ(y) f(y) 2 dμ(y))dμ(x). f o (x)e(φ 1 (y)) 2 dμ(y)dμ(x). f 2. K φ 2. f 2 <. Hence V φ is bounded Volterra composition operator.

3 Volterra composition operators 347 Theorem 1.2 Let V φ B(L 2 (μ)). Then Vφ f f o E(V f)oφ 1 Proof: Let f,g L 2 (μ). Consider Thus for μ-almost all x. Hence V φ f,g f,v φ g 1 1 f(x)v φ g(x)dμ(x) f(x) goφ(y)( (goφ)(y)dμ(y)dμ(x) 1 y f(x)dμ(x)dμ(y) goφ(y)(v f)(y)dμ(y) V f,c φ g Cφ V f,g Vφ f(x) (CφV f)(x) f o E((V f)oφ 1 (x)) V φ f f o E(V f)oφ 1. Theorem 1.3 Let V φ B(L 2 [, 1]). Then V φ is idempotent if and only if V φ is the zero operator. Proof : Suppose V φ is idempotent. Then Vφ 2f V φf. Taking f χ [,1], implies that φ(x) φ(t)dt φ(x), for all x. Differentiating, we get φ(φ(x))φ (x) φ (x) Suppose φ (x) for μ-almost all x E,μ(E) >. Then φ(φ(x)) 1 for μ-almost all x E. Next f(x) x for every x [, 1], we have (V φ f)v φ (V φ f)(x) φ(x) (φ(t)) 2 2 dt (φ(x))2. 2

4 348 A. Gupta and B. S. Komal On differentiating with respect to x, This yields that φ(x) 1 2, so that φ (x), which is contradiction. Further, suppose φ (x) a.e. Then φ(x) c for μ-almost all x [, 1]. we have (V φ f)(x) c and V φ (V φ f)(x) c 2. This implies that c or c 1. Now taking c 1 and f (x) x, again from (1) and (2) we get c 2 2 c3. i.e. 1 2, which is absurd. Hence c must be zero. Thus φ(x) for μ-almost every x [.1]. φ(x) Hence (V φ f)(x) f(x) for every x X and f L 2 ([, 1]). This proves that V φ. The converse is trivial. Theorem 1.4 The Volterra composition operator is never an identity operator. Proof: If possible, suppose a Volterra composition operator V φ for some φ is the identity operator, I. Then V φ f f for every f L 2 ([, 1]). taking f χ [,1], we have φ(x) f(t)dt f(x) for μ-almost all x [, 1] Hence from (1) Hence V φ I. 1 f(t)dt f(x) for all x [, 1], which is absurd. Theorem 1.5 Suppose φ(). Then σ(v φ ){}. Proof : If possible, suppose σ(v φ ) {}. Let λ is an eigen value of V φ. For, f,f L 2 (μ). (V φ f)(x) λf(x) Differentiating with respect to x, we get f(φ(y))dμ(y)λf(x) d x dx ( f(φ(y))dμ(y)) λf (x)

5 Volterra composition operators 349 f(φ(x) λf (x) f (x) 1 λ (foφ)(x) 1 λ f(φ(x)) From (1) f(). From (2) f () Also f (),... and so on f n (). This implies that f is zero function, which is a contradiction Hence σ(v φ ) {}. Example 1.6 Suppose (i) φ(x) ax, 1 >a>: x [o, 1]. (ii) φ(x) x n,n>1. Then σ(v φ ){}. Solution : (i) Suppose o λ is an eigen value of V φ Then f(ay)dy λf(x) (1) Put ay t, a dy dt x 1 i.e. f(t)dt λf(x) α Differentiating with respect to x, we get f(φ(x)) λf (x) From (1), f(), From (2), f ()... and so on f n () Hence σ(v φ ){}. (ii) If φ(x) x n we have f(φ(y)dy λf(x) Put y n t ny n 1 dy dt Put this value in (3), we have f (x) 1 λ f (φ(x)) (2) f(y n )dy λf(x) (3) dy 1 n t1/n 1 dt 1 f(t)t 1/n 1 dt λf(x) n Differentiating both sides with respect to x. We have 1 n f(x)x1/n 1 λf (x) (4)

6 35 A. Gupta and B. S. Komal From (3), f(). From (4), f ()... and so on f n () Hence, the result. References [1] A.Gupta and B.S.Komal, Composite integral operator on L 2 (μ), Pitman Lecture Notes in Mathematics series 377, 92-99,(1997). [2] A. Lambert, Normal extension of subnormal composition operators, Michigan Math. J., 35,443-45, (1988). [3] C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced mathematics CRC Press, New York (1995). [4] E. A. Nordgen, Composition operators on Hilbert spaces, Lecture notes in Math, 693, Springer -Verlag, New York, 37-63, (1978). [5] G. Sinnamon, Weighted Hardy and opial type inequalities J. Math. Anal. Appl. 16, , (1991). [6] J.A. Erdos, The commutant of the Voterra operator, Integral Equations and operator theory, 5, , (1982). [7] J.H. Shapiro, Composition operators and classical function theory, Springer- Verley, New York. (1993). [8] K.R. Parthasarathy, Introduction to probability and measure, Macmillion Limited, (1977). [9] P.R. Halmos and V.S. Sunder, Bounded integral operators on L 2 -spaces, Springer-Verlag, New York, (1978). [1] R.K. Singh and B.S. Komal, Composite integral operator on lp and its adjoint, Proc. Amer. Math. Soc. 7,21-25,(1978). [11] R.K. Singh and J.S. Manhas, Composition operators on function spaces, North Holland Mathematics studies 179, Elsevier sciences publishers Amsterdam, New York (1993). [12] R. Whitley, The spectrum of a Volterra composition operator, Integral equation and operator theory Vol. 1 (1997). [13] V.D. Stepanov, Weighted norm inequalities of Hardy type for a class of integral operators, J. London Math. Soc. (2) 5, 15-12, (1994).

7 Volterra composition operators 351 [14] V.D. Stepanov, Weighted inequalities for a class of Volterra convolution operators, J. London Math , (1992). [15] Yu. I. Lyubic, Composition of integration and substitution, Linear and complex Analysis, Problem Book, Springer Lect. Notes in Maths., 143, Berlin , (1984). Received: September, 21